I get a lot of fairly "technical" email from people like Istvan, Jason, or Ben. I have tried in the past to rewrite their ideas and information into something that doesn't read like a textbook. The down-side of that is I think I misinterpret some of their findings, or just flat out get it wrong!! The following is an idea I had to ensure a couple of things happen.
1. I wanted to ensure that people got credit correctly for their ideas.
2. I didn't want to run the risk of stating something that hadn't been said. Just like that kids game "telephone" where people whisper a message to the person next to them, and it goes down a line, and by the time it gets back to the first person, it's been altered, mangled, and just plain changed. I wanted to make sure that I got the INFORMATION out, not my understanding of the information.
3. I really believe that something someone says, will make someone else think of something else, which will make someone else think of something even different. (Now, THERE's a sentence!! :-) ) If everyone agrees, there could be a lot of "cross pollination" going on. This seems like a good way to get different ideas to a lot of people.
I'll try to give "background information" if an email needs it to be clearer.
I may also edit formating or content, but I will not "change" the info. For example, if a note talks about Lychrel Numbers, and also the fact that Istvan is going on vacation, I probably will delete the part about the vacation, but his discussion on Lychrel Numbers will be intact.
I will make every effort to go back and note an old email, if something new is found or if something is found to be wrong. I don't expect anyone to alter their style of writing in their email. I have used bits and pieces of email in my pages before, but this time I have tried to ask for more "formal" approval from everyone before I add their notes. If there is something anyone wants deleted, let me know!!
Note: Anything not in ITALICS are my comments, and was not part of the written email.
I don't know if this will be of any value or not, but I am going to try. :-)
So in reverse order of received date, here we go....
9/26/05 From: Jason Doucette To: Wade
Hi Wade,
On September 25, Sunday morning, 3:16am, my program completed the 18-digit set for the Most Delayed Palindromic Number quest. This set did not produce a new world record. Here is the program's output:
Solved all 18 digit numbers at Sun Sep 25 03:16:02 2005.
1,500,000,764 total numbers solve in 1 iteration.
6,328,923,305 total numbers solve in 2 iterations.
4,010,493,144 total numbers solve in 3 iterations.
4,516,639,086 total numbers solve in 4 iterations.
3,521,084,796 total numbers solve in 5 iterations.
3,063,902,965 total numbers solve in 6 iterations.
2,704,900,044 total numbers solve in 7 iterations.
2,533,614,778 total numbers solve in 8 iterations.
2,277,842,787 total numbers solve in 9 iterations.
1,934,167,473 total numbers solve in 10 iterations.
1,895,046,373 total numbers solve in 11 iterations.
1,632,602,785 total numbers solve in 12 iterations.
1,493,289,585 total numbers solve in 13 iterations.
1,323,522,629 total numbers solve in 14 iterations.
1,283,449,041 total numbers solve in 15 iterations.
1,121,799,008 total numbers solve in 16 iterations.
1,038,681,037 total numbers solve in 17 iterations.
914,234,922 total numbers solve in 18 iterations.
847,452,244 total numbers solve in 19 iterations.
774,047,924 total numbers solve in 20 iterations.
689,485,395 total numbers solve in 21 iterations.
641,222,470 total numbers solve in 22 iterations.
569,863,490 total numbers solve in 23 iterations.
529,356,696 total numbers solve in 24 iterations.
473,945,759 total numbers solve in 25 iterations.
433,634,237 total numbers solve in 26 iterations.
392,381,440 total numbers solve in 27 iterations.
359,624,144 total numbers solve in 28 iterations.
324,729,645 total numbers solve in 29 iterations.
296,780,172 total numbers solve in 30 iterations.
268,493,620 total numbers solve in 31 iterations.
243,979,250 total numbers solve in 32 iterations.
221,067,120 total numbers solve in 33 iterations.
201,964,310 total numbers solve in 34 iterations.
182,702,866 total numbers solve in 35 iterations.
166,934,119 total numbers solve in 36 iterations.
153,036,390 total numbers solve in 37 iterations.
138,637,320 total numbers solve in 38 iterations.
126,081,198 total numbers solve in 39 iterations.
114,597,327 total numbers solve in 40 iterations.
104,697,570 total numbers solve in 41 iterations.
95,720,030 total numbers solve in 42 iterations.
87,170,652 total numbers solve in 43 iterations.
79,518,356 total numbers solve in 44 iterations.
72,743,289 total numbers solve in 45 iterations.
66,150,591 total numbers solve in 46 iterations.
60,441,698 total numbers solve in 47 iterations.
54,903,581 total numbers solve in 48 iterations.
50,134,549 total numbers solve in 49 iterations.
45,723,123 total numbers solve in 50 iterations.
41,668,042 total numbers solve in 51 iterations.
37,954,950 total numbers solve in 52 iterations.
34,585,631 total numbers solve in 53 iterations.
31,506,829 total numbers solve in 54 iterations.
28,591,192 total numbers solve in 55 iterations.
26,153,920 total numbers solve in 56 iterations.
23,871,083 total numbers solve in 57 iterations.
21,872,970 total numbers solve in 58 iterations.
19,995,901 total numbers solve in 59 iterations.
18,257,738 total numbers solve in 60 iterations.
16,670,535 total numbers solve in 61 iterations.
15,132,809 total numbers solve in 62 iterations.
13,829,011 total numbers solve in 63 iterations.
12,587,072 total numbers solve in 64 iterations.
11,513,105 total numbers solve in 65 iterations.
10,482,532 total numbers solve in 66 iterations.
9,588,944 total numbers solve in 67 iterations.
8,743,573 total numbers solve in 68 iterations.
7,995,569 total numbers solve in 69 iterations.
7,316,081 total numbers solve in 70 iterations.
6,673,240 total numbers solve in 71 iterations.
6,075,315 total numbers solve in 72 iterations.
5,548,328 total numbers solve in 73 iterations.
5,044,984 total numbers solve in 74 iterations.
4,622,940 total numbers solve in 75 iterations.
4,191,786 total numbers solve in 76 iterations.
3,847,062 total numbers solve in 77 iterations.
3,488,591 total numbers solve in 78 iterations.
3,191,717 total numbers solve in 79 iterations.
2,930,163 total numbers solve in 80 iterations.
2,669,240 total numbers solve in 81 iterations.
2,453,755 total numbers solve in 82 iterations.
2,248,628 total numbers solve in 83 iterations.
2,071,593 total numbers solve in 84 iterations.
1,882,970 total numbers solve in 85 iterations.
1,734,406 total numbers solve in 86 iterations.
1,582,680 total numbers solve in 87 iterations.
1,443,442 total numbers solve in 88 iterations.
1,313,840 total numbers solve in 89 iterations.
1,185,189 total numbers solve in 90 iterations.
1,084,951 total numbers solve in 91 iterations.
990,267 total numbers solve in 92 iterations.
897,631 total numbers solve in 93 iterations.
816,091 total numbers solve in 94 iterations.
764,544 total numbers solve in 95 iterations.
689,078 total numbers solve in 96 iterations.
639,099 total numbers solve in 97 iterations.
572,585 total numbers solve in 98 iterations.
528,950 total numbers solve in 99 iterations.
481,999 total numbers solve in 100 iterations.
442,485 total numbers solve in 101 iterations.
413,232 total numbers solve in 102 iterations.
372,998 total numbers solve in 103 iterations.
342,277 total numbers solve in 104 iterations.
309,368 total numbers solve in 105 iterations.
282,604 total numbers solve in 106 iterations.
259,478 total numbers solve in 107 iterations.
233,588 total numbers solve in 108 iterations.
214,553 total numbers solve in 109 iterations.
201,651 total numbers solve in 110 iterations.
182,911 total numbers solve in 111 iterations.
163,718 total numbers solve in 112 iterations.
150,673 total numbers solve in 113 iterations.
136,388 total numbers solve in 114 iterations.
131,595 total numbers solve in 115 iterations.
115,050 total numbers solve in 116 iterations.
107,648 total numbers solve in 117 iterations.
97,021 total numbers solve in 118 iterations.
89,065 total numbers solve in 119 iterations.
79,227 total numbers solve in 120 iterations.
74,074 total numbers solve in 121 iterations.
61,361 total numbers solve in 122 iterations.
61,620 total numbers solve in 123 iterations.
53,464 total numbers solve in 124 iterations.
49,566 total numbers solve in 125 iterations.
47,399 total numbers solve in 126 iterations.
41,612 total numbers solve in 127 iterations.
38,980 total numbers solve in 128 iterations.
36,131 total numbers solve in 129 iterations.
33,727 total numbers solve in 130 iterations.
27,269 total numbers solve in 131 iterations.
25,223 total numbers solve in 132 iterations.
21,282 total numbers solve in 133 iterations.
20,896 total numbers solve in 134 iterations.
20,450 total numbers solve in 135 iterations.
18,831 total numbers solve in 136 iterations.
15,560 total numbers solve in 137 iterations.
14,969 total numbers solve in 138 iterations.
14,756 total numbers solve in 139 iterations.
13,969 total numbers solve in 140 iterations.
11,349 total numbers solve in 141 iterations.
10,441 total numbers solve in 142 iterations.
8,148 total numbers solve in 143 iterations.
8,123 total numbers solve in 144 iterations.
7,885 total numbers solve in 145 iterations.
7,464 total numbers solve in 146 iterations.
5,951 total numbers solve in 147 iterations.
5,682 total numbers solve in 148 iterations.
4,560 total numbers solve in 149 iterations.
3,318 total numbers solve in 150 iterations.
3,121 total numbers solve in 151 iterations.
2,866 total numbers solve in 152 iterations.
2,274 total numbers solve in 153 iterations.
2,077 total numbers solve in 154 iterations.
2,375 total numbers solve in 155 iterations.
2,134 total numbers solve in 156 iterations.
1,803 total numbers solve in 157 iterations.
1,228 total numbers solve in 158 iterations.
916 total numbers solve in 159 iterations.
692 total numbers solve in 160 iterations.
1,747 total numbers solve in 161 iterations.
1,309 total numbers solve in 162 iterations.
1,480 total numbers solve in 163 iterations.
1,568 total numbers solve in 164 iterations.
921 total numbers solve in 165 iterations.
649 total numbers solve in 166 iterations.
631 total numbers solve in 167 iterations.
500 total numbers solve in 168 iterations.
545 total numbers solve in 169 iterations.
1,159 total numbers solve in 170 iterations.
788 total numbers solve in 171 iterations.
996 total numbers solve in 172 iterations.
614 total numbers solve in 173 iterations.
294 total numbers solve in 174 iterations.
171 total numbers solve in 175 iterations.
1,149 total numbers solve in 176 iterations.
1,552 total numbers solve in 177 iterations.
1,341 total numbers solve in 178 iterations.
779 total numbers solve in 179 iterations.
958 total numbers solve in 180 iterations.
410 total numbers solve in 181 iterations.
767 total numbers solve in 182 iterations.
683 total numbers solve in 183 iterations.
367 total numbers solve in 184 iterations.
376 total numbers solve in 185 iterations.
215 total numbers solve in 186 iterations.
302 total numbers solve in 187 iterations.
176 total numbers solve in 188 iterations.
156 total numbers solve in 189 iterations.
129 total numbers solve in 190 iterations.
150 total numbers solve in 191 iterations.
100 total numbers solve in 192 iterations.
41 total numbers solve in 193 iterations.
17 total numbers solve in 194 iterations.
14 total numbers solve in 195 iterations.
9 total numbers solve in 196 iterations.
5 total numbers solve in 197 iterations.
14 total numbers solve in 198 iterations.
7 total numbers solve in 199 iterations.
4 total numbers solve in 200 iterations.
7 total numbers solve in 201 iterations.
195 total numbers solve in 202 iterations.
106 total numbers solve in 203 iterations.
273 total numbers solve in 204 iterations.
171 total numbers solve in 205 iterations.
100 total numbers solve in 206 iterations.
44 total numbers solve in 207 iterations.
24 total numbers solve in 208 iterations.
49 total numbers solve in 209 iterations.
21 total numbers solve in 210 iterations.
10 total numbers solve in 211 iterations.
7 total numbers solve in 212 iterations.
64 total numbers solve in 213 iterations.
32 total numbers solve in 214 iterations.
16 total numbers solve in 215 iterations.
1 total numbers solve in 216 iterations.
0 total numbers solve in 217 iterations.
32 total numbers solve in 218 iterations.
24 total numbers solve in 219 iterations.
9 total numbers solve in 220 iterations.
3 total numbers solve in 221 iterations.
1 total numbers solve in 222 iterations.
0 total numbers solve in 223 iterations.
0 total numbers solve in 224 iterations.
0 total numbers solve in 225 iterations.
0 total numbers solve in 226 iterations.
96 total numbers solve in 227 iterations.
120 total numbers solve in 228 iterations.
337 total numbers solve in 229 iterations.
313 total numbers solve in 230 iterations.
396 total numbers solve in 231 iterations.
171 total numbers solve in 232 iterations.
81 total numbers solve in 233 iterations.
14 total numbers solve in 234 iterations.
2 total numbers solve in 235 iterations.
1 total numbers solve in 236 iterations.
440,086,795,650 of 492,523,328,187 total numbers did not solve out (89.35%).
52,436,532,537 numbers have solved so far.
NOTE that the above two lines represent ALL numbers tested from 1-digit all the way through to 18-digit. I have computed the individual information per digit set, using Excel, since my program does not natively produce this output, and I will work to upload these to my site sometime soon. I apologize that I am extremely busy with a project at the moment, so this may be a little slow. I do have a table of the % of Lychrels per each digit set, computed, though, which is available off of my website:
|
1-digit numbers 2-digit numbers 3-digit numbers 4-digit numbers 5-digit numbers 6-digit numbers 7-digit numbers 8-digit numbers 9-digit numbers 10-digit numbers 11-digit numbers 12-digit numbers 13-digit numbers 14-digit numbers 15-digit numbers 16-digit numbers 17-digit numbers 18-digit numbers |
0.00% 0.00% 1.67% 3.51% 7.25% 14.45% 22.17% 31.30% 40.42% 49.61% 57.82% 65.44% 71.64% 77.17% 81.41% 85.22% 88.03% 90.55% |
More output from my program:
Processed all 18 digit numbers in 21,199,664 seconds = 5,888 hours = 245 days = 0.67 years.
Total processing = 64,427,388 seconds = 17,896 hours = 745 days = 2.04 years.
Number of 18 Digit Numbers Checked = 305,704,134,738
Number of Numbers Checked = 492,523,328,187
Note that I only checked about 305 billion (305,704,134,738) number of 18-digit numbers, not all 900,000,000,000,000,000 of them. Thus the program was running almost 3,000,000 times faster with the optimization. It would have taken just over 6,000,000 years to perform this without the optimization, using the same computer system.
The 19-digit set has commenced. There are 3,057,041,347,380 number of 19 digit numbers to check (exactly 10x the amount of the 18-digit set) -- over 3 trillion numbers! Of course, this is much better than checking every 19 digit number, of which there are 9,000,000,000,000,000,000. So the program's optimization helps this greatly...
Take care,
Jason Doucette
12/15/04 From: Jason Doucette To: Wade
Hi Wade,
My program finished computing all 17-digit numbers yesterday. Here's the program's output:
600,000,508 total numbers solve in 1 iteration.
2,958,708,982 total numbers solve in 2 iterations.
1,919,554,008 total numbers solve in 3 iterations.
2,076,217,013 total numbers solve in 4 iterations.
1,582,873,888 total numbers solve in 5 iterations.
1,361,417,719 total numbers solve in 6 iterations.
1,205,930,592 total numbers solve in 7 iterations.
1,131,200,765 total numbers solve in 8 iterations.
1,009,703,707 total numbers solve in 9 iterations.
857,728,968 total numbers solve in 10 iterations.
842,779,933 total numbers solve in 11 iterations.
728,316,192 total numbers solve in 12 iterations.
665,387,178 total numbers solve in 13 iterations.
588,048,095 total numbers solve in 14 iterations.
568,413,029 total numbers solve in 15 iterations.
498,107,075 total numbers solve in 16 iterations.
458,852,289 total numbers solve in 17 iterations.
401,255,825 total numbers solve in 18 iterations.
371,801,697 total numbers solve in 19 iterations.
341,450,810 total numbers solve in 20 iterations.
304,923,968 total numbers solve in 21 iterations.
282,497,625 total numbers solve in 22 iterations.
249,799,220 total numbers solve in 23 iterations.
232,263,252 total numbers solve in 24 iterations.
208,913,316 total numbers solve in 25 iterations.
191,183,987 total numbers solve in 26 iterations.
172,388,263 total numbers solve in 27 iterations.
157,720,874 total numbers solve in 28 iterations.
142,709,155 total numbers solve in 29 iterations.
130,613,100 total numbers solve in 30 iterations.
117,934,237 total numbers solve in 31 iterations.
106,947,215 total numbers solve in 32 iterations.
96,872,267 total numbers solve in 33 iterations.
88,575,292 total numbers solve in 34 iterations.
80,207,267 total numbers solve in 35 iterations.
73,169,871 total numbers solve in 36 iterations.
66,952,405 total numbers solve in 37 iterations.
60,764,061 total numbers solve in 38 iterations.
55,254,746 total numbers solve in 39 iterations.
50,202,558 total numbers solve in 40 iterations.
45,802,571 total numbers solve in 41 iterations.
41,851,417 total numbers solve in 42 iterations.
38,175,540 total numbers solve in 43 iterations.
34,797,095 total numbers solve in 44 iterations.
31,828,812 total numbers solve in 45 iterations.
28,952,076 total numbers solve in 46 iterations.
26,386,934 total numbers solve in 47 iterations.
24,010,256 total numbers solve in 48 iterations.
21,966,684 total numbers solve in 49 iterations.
20,019,171 total numbers solve in 50 iterations.
18,207,020 total numbers solve in 51 iterations.
16,628,726 total numbers solve in 52 iterations.
15,174,276 total numbers solve in 53 iterations.
13,789,270 total numbers solve in 54 iterations.
12,506,400 total numbers solve in 55 iterations.
11,444,843 total numbers solve in 56 iterations.
10,385,254 total numbers solve in 57 iterations.
9,559,855 total numbers solve in 58 iterations.
8,720,365 total numbers solve in 59 iterations.
7,992,353 total numbers solve in 60 iterations.
7,305,403 total numbers solve in 61 iterations.
6,617,289 total numbers solve in 62 iterations.
6,059,671 total numbers solve in 63 iterations.
5,485,722 total numbers solve in 64 iterations.
5,018,477 total numbers solve in 65 iterations.
4,578,738 total numbers solve in 66 iterations.
4,219,822 total numbers solve in 67 iterations.
3,796,991 total numbers solve in 68 iterations.
3,473,573 total numbers solve in 69 iterations.
3,190,167 total numbers solve in 70 iterations.
2,910,917 total numbers solve in 71 iterations.
2,648,213 total numbers solve in 72 iterations.
2,424,255 total numbers solve in 73 iterations.
2,202,061 total numbers solve in 74 iterations.
2,017,044 total numbers solve in 75 iterations.
1,833,125 total numbers solve in 76 iterations.
1,676,346 total numbers solve in 77 iterations.
1,511,332 total numbers solve in 78 iterations.
1,378,277 total numbers solve in 79 iterations.
1,268,288 total numbers solve in 80 iterations.
1,174,437 total numbers solve in 81 iterations.
1,085,807 total numbers solve in 82 iterations.
981,780 total numbers solve in 83 iterations.
894,708 total numbers solve in 84 iterations.
821,457 total numbers solve in 85 iterations.
766,852 total numbers solve in 86 iterations.
695,313 total numbers solve in 87 iterations.
624,326 total numbers solve in 88 iterations.
571,780 total numbers solve in 89 iterations.
526,196 total numbers solve in 90 iterations.
484,663 total numbers solve in 91 iterations.
439,503 total numbers solve in 92 iterations.
392,517 total numbers solve in 93 iterations.
358,250 total numbers solve in 94 iterations.
332,527 total numbers solve in 95 iterations.
294,918 total numbers solve in 96 iterations.
273,627 total numbers solve in 97 iterations.
249,528 total numbers solve in 98 iterations.
230,571 total numbers solve in 99 iterations.
205,565 total numbers solve in 100 iterations.
183,895 total numbers solve in 101 iterations.
178,726 total numbers solve in 102 iterations.
161,305 total numbers solve in 103 iterations.
155,751 total numbers solve in 104 iterations.
139,606 total numbers solve in 105 iterations.
121,852 total numbers solve in 106 iterations.
111,928 total numbers solve in 107 iterations.
100,622 total numbers solve in 108 iterations.
93,732 total numbers solve in 109 iterations.
88,931 total numbers solve in 110 iterations.
79,989 total numbers solve in 111 iterations.
69,628 total numbers solve in 112 iterations.
65,136 total numbers solve in 113 iterations.
63,315 total numbers solve in 114 iterations.
59,019 total numbers solve in 115 iterations.
52,708 total numbers solve in 116 iterations.
45,540 total numbers solve in 117 iterations.
40,907 total numbers solve in 118 iterations.
37,679 total numbers solve in 119 iterations.
34,330 total numbers solve in 120 iterations.
34,479 total numbers solve in 121 iterations.
28,217 total numbers solve in 122 iterations.
25,763 total numbers solve in 123 iterations.
24,698 total numbers solve in 124 iterations.
22,148 total numbers solve in 125 iterations.
18,751 total numbers solve in 126 iterations.
17,395 total numbers solve in 127 iterations.
17,454 total numbers solve in 128 iterations.
16,166 total numbers solve in 129 iterations.
14,353 total numbers solve in 130 iterations.
12,478 total numbers solve in 131 iterations.
11,565 total numbers solve in 132 iterations.
9,206 total numbers solve in 133 iterations.
8,328 total numbers solve in 134 iterations.
8,451 total numbers solve in 135 iterations.
8,304 total numbers solve in 136 iterations.
7,124 total numbers solve in 137 iterations.
6,830 total numbers solve in 138 iterations.
7,081 total numbers solve in 139 iterations.
4,837 total numbers solve in 140 iterations.
3,906 total numbers solve in 141 iterations.
4,884 total numbers solve in 142 iterations.
3,901 total numbers solve in 143 iterations.
3,726 total numbers solve in 144 iterations.
2,427 total numbers solve in 145 iterations.
2,189 total numbers solve in 146 iterations.
2,823 total numbers solve in 147 iterations.
3,081 total numbers solve in 148 iterations.
2,272 total numbers solve in 149 iterations.
1,818 total numbers solve in 150 iterations.
1,254 total numbers solve in 151 iterations.
1,350 total numbers solve in 152 iterations.
775 total numbers solve in 153 iterations.
804 total numbers solve in 154 iterations.
508 total numbers solve in 155 iterations.
420 total numbers solve in 156 iterations.
508 total numbers solve in 157 iterations.
456 total numbers solve in 158 iterations.
514 total numbers solve in 159 iterations.
461 total numbers solve in 160 iterations.
486 total numbers solve in 161 iterations.
486 total numbers solve in 162 iterations.
671 total numbers solve in 163 iterations.
482 total numbers solve in 164 iterations.
413 total numbers solve in 165 iterations.
429 total numbers solve in 166 iterations.
362 total numbers solve in 167 iterations.
175 total numbers solve in 168 iterations.
144 total numbers solve in 169 iterations.
660 total numbers solve in 170 iterations.
369 total numbers solve in 171 iterations.
366 total numbers solve in 172 iterations.
182 total numbers solve in 173 iterations.
64 total numbers solve in 174 iterations.
22 total numbers solve in 175 iterations.
183 total numbers solve in 176 iterations.
626 total numbers solve in 177 iterations.
697 total numbers solve in 178 iterations.
376 total numbers solve in 179 iterations.
459 total numbers solve in 180 iterations.
284 total numbers solve in 181 iterations.
241 total numbers solve in 182 iterations.
206 total numbers solve in 183 iterations.
167 total numbers solve in 184 iterations.
298 total numbers solve in 185 iterations.
170 total numbers solve in 186 iterations.
86 total numbers solve in 187 iterations.
62 total numbers solve in 188 iterations.
41 total numbers solve in 189 iterations.
31 total numbers solve in 190 iterations.
89 total numbers solve in 191 iterations.
63 total numbers solve in 192 iterations.
33 total numbers solve in 193 iterations.
17 total numbers solve in 194 iterations.
8 total numbers solve in 195 iterations.
5 total numbers solve in 196 iterations.
2 total numbers solve in 197 iterations.
12 total numbers solve in 198 iterations.
7 total numbers solve in 199 iterations.
4 total numbers solve in 200 iterations.
3 total numbers solve in 201 iterations.
0 total numbers solve in 202 iterations.
0 total numbers solve in 203 iterations.
0 total numbers solve in 204 iterations.
0 total numbers solve in 205 iterations.
16 total numbers solve in 206 iterations.
12 total numbers solve in 207 iterations.
24 total numbers solve in 208 iterations.
11 total numbers solve in 209 iterations.
4 total numbers solve in 210 iterations.
0 total numbers solve in 211 iterations.
0 total numbers solve in 212 iterations.
0 total numbers solve in 213 iterations.
0 total numbers solve in 214 iterations.
0 total numbers solve in 215 iterations.
0 total numbers solve in 216 iterations.
0 total numbers solve in 217 iterations.
0 total numbers solve in 218 iterations.
0 total numbers solve in 219 iterations.
0 total numbers solve in 220 iterations.
0 total numbers solve in 221 iterations.
0 total numbers solve in 222 iterations.
0 total numbers solve in 223 iterations.
0 total numbers solve in 224 iterations.
0 total numbers solve in 225 iterations.
0 total numbers solve in 226 iterations.
0 total numbers solve in 227 iterations.
0 total numbers solve in 228 iterations.
72 total numbers solve in 229 iterations.
201 total numbers solve in 230 iterations.
326 total numbers solve in 231 iterations.
149 total numbers solve in 232 iterations.
81 total numbers solve in 233 iterations.
14 total numbers solve in 234 iterations.
2 total numbers solve in 235 iterations.
1 total numbers solve in 236 iterations.
163,260,096,845 of 186,819,193,449 total numbers did not solve out (87.39%). Again, this percentage is increasing for each new number length:
17-digit numbers: 87.39%
16-digit numbers: 83.42%
15-digit numbers: 80.46%
14-digit numbers: 74.56%
13-digit numbers: 70.29%
12-digit numbers: 61.89%
11-digit numbers: 56.10%
10-digit numbers: 45.43%
9-digit numbers: 38.59%
8-digit numbers: 27.27%
7-digit numbers: 20.69%
6-digit numbers: 11.48%
5-digit numbers: 6.63%
4-digit numbers: 2.73%
3-digit numbers: 1.45%
2-digit numbers: 0.00%
1-digit numbers: 0.00%
Again, I will explain: Due to the optimizations in my algorithm, it does not iteratively check each number. My algorithm determines which numbers can be eliminated from the search and still maintain 100% accurate results. As a result, my program actually only checked 186,819,193,449 total numbers, not 99,999,999,999,999,999 numbers, which is a significant optimization. It would have taken over 700,000 years, on the same computer, to compute all these numbers without this optimization.
P.S. Right now my program is only being run 16 hours / day on my 1.83 GHz Athlon. If you know someone with more CPU power (either more hours to dedicate or a faster processor) who wishes to continue this quest, then let me know. I will be glad to hand over my program. I can still maintain updates on my website when the program reports records. Right now, due to my heavy involvement of my current graphics project, it is likely I will be using a lot of CPU power for testing/renderer purposes, and this quest will be slowed as a result. Let me know...
Take care,
---------
Jason Doucette
http://www.jasondoucette.com/
9/16/03 From: Vaughn Suite To: Wade
Wade,
I did some work on checking the number of numbers that form palindromes without carries and with carries... This helps demonstrate the (im)probability of ever finding a palindrome as the number of digits increases.
I am enclosing a program (pcarry0.exe) which performs a single reverse and add on all numbers beginning with 1 digit numbers and reports those which form a palindrome. Strictly speaking, this program should operate in exponential time, each extra digit taking 10 times as long as the previous, [version 1 took 3 seconds to reach 3 digits, 31 seconds to 4 digits and 316 seconds to 5 digits on my k6-2], however there is some logical optimization which makes it faster than that, so this version took 8 seconds to 4 digits, 47 seconds to 5 digits, 474 seconds to 6 and 3340 to 7. The point is, it is still very slow.
Kruppa's data at http://home.cfl.rr.com/p196/kruppa.txt demonstrates those numbers which form a palindrome after a carry.
However, given the entire predictability of the numbers, there is no need for either a program to check whether each n digit number forms a palindrome.
For odd n>=3
A B...C D C'...B'A' will reverse and add to a palindrome of the same length if A+A', B+B', C+C' and D+D are less than 9 (no carries).
A B...C D C'...B'A' will reverse and add to a palindrome of length n+1 if A+A'=11, B+B' until C+C'=0 or 11, and D=0
For even n>=2
A B...C C'...B'A' will reverse and add to a palindrome of the same length if A+A', B+B' and C+C are less than 9 (no carries).
A B...C C'...B'A' will reverse and add to a palindrome of length n+1 if if A+A'=11, B+B' until C+C'=0 or 11.
You will see that there are 9*10^(n-1) possible n digit numbers. Of these, if n is odd, there are 225*55^((n-3)/2) numbers that form a palindrome after a single reverse and add, and 8*9^((n-3)/2) numbers that form a palindrome after a single reverse and add. If n is even, these figures are 45*55^((n-2)/2) and 8*9^((n-2)/2) respectively.
As n increases by 2, there are 100 times as many possible n digit numbers but only 55 times as many which form palindromes without carry, and 9 times as many which form palindromes with a carry.
Now, on your wish list page, you ask about the number of carries in numbers which do not form palindromes. But this is not as important as a something actually in kruppa's data, but masked by how it is misrepresented.
Look again at the carry at the end of the first line. A two digit number 29 produces a 3 digit number, 121. The carry is not 011 as shown, but 11. The carry should be represented as the same length as the number being reversed and added.
Strip all the leading 0s from the list and the pattern becomes obvious. The carries in a number that produces a palindrome after reverse and add must be palindromic. ALL palindromes produced by a carry must have a carry in the first as well as last digit. Furthermore, the 2nd and 2nd-to-last carry must both be the same: 0 or 1, and so on for the 3rd and 3rd-to-last. If n is odd, then the middle carry must be 0; furthermore, any carry into the middle digit must be balanced by the value of the next digit, so the middle digit must be 0 so as not to affect the balance...
THE CARRY IS PALINDROMIC.
Now, it is so simple to prove from permutations/combinations that the above equations are correct. Using these equations, (pcarry1) shows the number of n digit numbers that form a palindrome after 1 reverse and add as n increases. Since it is a simple calculation, the answers appear instantaneously.
It is clear that there are quadrillions upon quadrillions of 120 million digit numbers that form a palindrome after a single reverse and add. How many exactly, you may ask. There are 9x10^119999999 different 120 million digit numbers. But the Windows calculator does not reach 10^325 for a 325 digit number. Nor does my OpenOffice.org Spreadsheet.
(pcarry2) uses (simple) advanced mathematics (as per pcarry1) to calculate the improbability of a 120 million digit number resolving into a palindrome.
Now, to distress ourselves even further by the magnitude of the task the palindrome quest poses, consider that by not forming a palindrome after 1 addition 196 is not one of the (225 + 8) 3 digit numbers that produce a palindrome after 1 reverse and add; and it produces 887 which is also not one. 1675 is not one of the (2475+8) 4 digit numbers which produces a palindrome after 1 add. And so on. The 120 million digit number yielded by the palindrome quest may not likely be one of the 10^104421760 (10 to the power of 104 million...) other 120 million digit numbers which actually yield a palindrome after reverse and add.
The improbability of the task is daunting!
Scientists count a 1 in 1000 chance as unlikely. This is why many modern scientists/mathematicians discount the probability of evolution having occurred.
[Quote] http://www.icr.org/pubs/imp/imp-073.htm Astro-physicists estimate that there are no more than 10^80 infinitesimal "particles" in the universe, and that the age of the universe in its present form is no greater than 10^18 seconds (30 billion years). Assuming each particle can participate in a thousand billion (10^12) different events every second (this is impossibly high, of course), then the greatest number of events that could ever happen (or trials that could ever be made) in all the universe throughout its entire history is only 10^80 x 10^18 x 10^12, or 10^110 (most authorities would make this figure much lower, about 10^50). Any event with a probability of less than one chance in 10^110, therefore, cannot occur. Its probability becomes zero, at least in our known universe.
Thus, the above-suggested ordered arrangement of 100 components has a zero probability. It could never happen by chance. Since every single living cell is infinitely more complex and ordered than this, it is impossible that even the simplest form of life could ever have originated by chance. Even the simplest replicating protein molecule that could be imagined has been shown by Golay1 to have a probability of one in 10^450. Salisbury2 calculates the probability of a typical DNA chain to be one in 10^600. [end quote]
The entire improbability of the 120 million digit set, or any other, eventually yielding a palindrome is striking.
We are not going anywhere.
The value of the comparison of software products, in my mind, is to see who can go nowhere fastest!!!
Unfortunately, with the exponential increase in time with linear increases in digit length, we must really plan to be here for the long haul.
But if it solves, it would be great, wouldn't it?
Cheers
Vaughn Suite
4/17/03 From: Jason Doucette To: Wade
Hello Wade,
My program finished computing all 16-digit numbers yesterday. Here are the results:
150,000,380 total numbers solve in 1 iterations.
523,549,055 total numbers solve in 2 iterations.
338,298,125 total numbers solve in 3 iterations.
376,841,978 total numbers solve in 4 iterations.
294,739,581 total numbers solve in 5 iterations.
256,144,958 total numbers solve in 6 iterations.
223,672,739 total numbers solve in 7 iterations.
207,520,576 total numbers solve in 8 iterations.
186,170,280 total numbers solve in 9 iterations.
157,224,107 total numbers solve in 10 iterations.
153,189,479 total numbers solve in 11 iterations.
131,815,779 total numbers solve in 12 iterations.
119,825,789 total numbers solve in 13 iterations.
105,887,339 total numbers solve in 14 iterations.
102,110,338 total numbers solve in 15 iterations.
89,252,268 total numbers solve in 16 iterations.
82,216,976 total numbers solve in 17 iterations.
72,384,260 total numbers solve in 18 iterations.
66,778,611 total numbers solve in 19 iterations.
60,924,158 total numbers solve in 20 iterations.
54,207,919 total numbers solve in 21 iterations.
50,272,062 total numbers solve in 22 iterations.
44,672,170 total numbers solve in 23 iterations.
41,394,309 total numbers solve in 24 iterations.
37,065,728 total numbers solve in 25 iterations.
33,887,347 total numbers solve in 26 iterations.
30,573,681 total numbers solve in 27 iterations.
27,982,034 total numbers solve in 28 iterations.
25,207,538 total numbers solve in 29 iterations.
23,036,661 total numbers solve in 30 iterations.
20,811,375 total numbers solve in 31 iterations.
18,929,337 total numbers solve in 32 iterations.
17,137,772 total numbers solve in 33 iterations.
15,650,249 total numbers solve in 34 iterations.
14,145,672 total numbers solve in 35 iterations.
12,922,968 total numbers solve in 36 iterations.
11,829,510 total numbers solve in 37 iterations.
10,686,954 total numbers solve in 38 iterations.
9,691,745 total numbers solve in 39 iterations.
8,794,855 total numbers solve in 40 iterations.
8,012,564 total numbers solve in 41 iterations.
7,344,186 total numbers solve in 42 iterations.
6,676,480 total numbers solve in 43 iterations.
6,091,264 total numbers solve in 44 iterations.
5,564,789 total numbers solve in 45 iterations.
5,066,727 total numbers solve in 46 iterations.
4,624,185 total numbers solve in 47 iterations.
4,236,674 total numbers solve in 48 iterations.
3,863,637 total numbers solve in 49 iterations.
3,509,079 total numbers solve in 50 iterations.
3,193,644 total numbers solve in 51 iterations.
2,902,885 total numbers solve in 52 iterations.
2,656,163 total numbers solve in 53 iterations.
2,423,390 total numbers solve in 54 iterations.
2,183,513 total numbers solve in 55 iterations.
2,002,224 total numbers solve in 56 iterations.
1,819,515 total numbers solve in 57 iterations.
1,677,058 total numbers solve in 58 iterations.
1,533,842 total numbers solve in 59 iterations.
1,395,594 total numbers solve in 60 iterations.
1,285,200 total numbers solve in 61 iterations.
1,161,394 total numbers solve in 62 iterations.
1,069,859 total numbers solve in 63 iterations.
969,013 total numbers solve in 64 iterations.
874,338 total numbers solve in 65 iterations.
801,382 total numbers solve in 66 iterations.
740,722 total numbers solve in 67 iterations.
673,393 total numbers solve in 68 iterations.
604,358 total numbers solve in 69 iterations.
546,369 total numbers solve in 70 iterations.
506,106 total numbers solve in 71 iterations.
465,959 total numbers solve in 72 iterations.
424,910 total numbers solve in 73 iterations.
388,194 total numbers solve in 74 iterations.
348,331 total numbers solve in 75 iterations.
319,420 total numbers solve in 76 iterations.
292,534 total numbers solve in 77 iterations.
267,052 total numbers solve in 78 iterations.
243,189 total numbers solve in 79 iterations.
221,094 total numbers solve in 80 iterations.
202,358 total numbers solve in 81 iterations.
182,826 total numbers solve in 82 iterations.
167,185 total numbers solve in 83 iterations.
151,345 total numbers solve in 84 iterations.
138,037 total numbers solve in 85 iterations.
133,124 total numbers solve in 86 iterations.
124,007 total numbers solve in 87 iterations.
108,733 total numbers solve in 88 iterations.
99,470 total numbers solve in 89 iterations.
91,022 total numbers solve in 90 iterations.
89,384 total numbers solve in 91 iterations.
77,201 total numbers solve in 92 iterations.
69,448 total numbers solve in 93 iterations.
62,680 total numbers solve in 94 iterations.
58,059 total numbers solve in 95 iterations.
51,953 total numbers solve in 96 iterations.
48,653 total numbers solve in 97 iterations.
43,647 total numbers solve in 98 iterations.
39,631 total numbers solve in 99 iterations.
35,819 total numbers solve in 100 iterations.
32,280 total numbers solve in 101 iterations.
29,720 total numbers solve in 102 iterations.
27,285 total numbers solve in 103 iterations.
26,999 total numbers solve in 104 iterations.
23,922 total numbers solve in 105 iterations.
23,451 total numbers solve in 106 iterations.
21,410 total numbers solve in 107 iterations.
18,425 total numbers solve in 108 iterations.
15,785 total numbers solve in 109 iterations.
16,303 total numbers solve in 110 iterations.
13,686 total numbers solve in 111 iterations.
12,966 total numbers solve in 112 iterations.
11,249 total numbers solve in 113 iterations.
10,686 total numbers solve in 114 iterations.
9,596 total numbers solve in 115 iterations.
9,333 total numbers solve in 116 iterations.
7,956 total numbers solve in 117 iterations.
7,977 total numbers solve in 118 iterations.
6,777 total numbers solve in 119 iterations.
6,492 total numbers solve in 120 iterations.
6,515 total numbers solve in 121 iterations.
5,938 total numbers solve in 122 iterations.
5,805 total numbers solve in 123 iterations.
4,823 total numbers solve in 124 iterations.
4,109 total numbers solve in 125 iterations.
3,639 total numbers solve in 126 iterations.
2,998 total numbers solve in 127 iterations.
2,331 total numbers solve in 128 iterations.
2,192 total numbers solve in 129 iterations.
1,455 total numbers solve in 130 iterations.
1,767 total numbers solve in 131 iterations.
1,358 total numbers solve in 132 iterations.
1,244 total numbers solve in 133 iterations.
1,431 total numbers solve in 134 iterations.
1,475 total numbers solve in 135 iterations.
1,383 total numbers solve in 136 iterations.
1,254 total numbers solve in 137 iterations.
1,452 total numbers solve in 138 iterations.
1,653 total numbers solve in 139 iterations.
1,289 total numbers solve in 140 iterations.
729 total numbers solve in 141 iterations.
628 total numbers solve in 142 iterations.
584 total numbers solve in 143 iterations.
581 total numbers solve in 144 iterations.
529 total numbers solve in 145 iterations.
397 total numbers solve in 146 iterations.
176 total numbers solve in 147 iterations.
111 total numbers solve in 148 iterations.
157 total numbers solve in 149 iterations.
195 total numbers solve in 150 iterations.
191 total numbers solve in 151 iterations.
232 total numbers solve in 152 iterations.
117 total numbers solve in 153 iterations.
180 total numbers solve in 154 iterations.
99 total numbers solve in 155 iterations.
74 total numbers solve in 156 iterations.
32 total numbers solve in 157 iterations.
13 total numbers solve in 158 iterations.
5 total numbers solve in 159 iterations.
0 total numbers solve in 160 iterations - the only one missing!
160 total numbers solve in 161 iterations.
88 total numbers solve in 162 iterations.
235 total numbers solve in 163 iterations.
145 total numbers solve in 164 iterations.
81 total numbers solve in 165 iterations.
33 total numbers solve in 166 iterations.
157 total numbers solve in 167 iterations.
84 total numbers solve in 168 iterations.
92 total numbers solve in 169 iterations.
46 total numbers solve in 170 iterations.
67 total numbers solve in 171 iterations.
151 total numbers solve in 172 iterations.
90 total numbers solve in 173 iterations.
40 total numbers solve in 174 iterations.
18 total numbers solve in 175 iterations.
10 total numbers solve in 176 iterations.
128 total numbers solve in 177 iterations.
379 total numbers solve in 178 iterations.
192 total numbers solve in 179 iterations.
158 total numbers solve in 180 iterations.
111 total numbers solve in 181 iterations.
48 total numbers solve in 182 iterations.
123 total numbers solve in 183 iterations.
80 total numbers solve in 184 iterations.
100 total numbers solve in 185 iterations.
64 total numbers solve in 186 iterations.
34 total numbers solve in 187 iterations.
30 total numbers solve in 188 iterations.
17 total numbers solve in 189 iterations.
9 total numbers solve in 190 iterations.
5 total numbers solve in 191 iterations.
2 total numbers solve in 192 iterations.
18 total numbers solve in 193 iterations.
6 total numbers solve in 194 iterations.
7 total numbers solve in 195 iterations.
5 total numbers solve in 196 iterations.
2 total numbers solve in 197 iterations.
12 total numbers solve in 198 iterations.
7 total numbers solve in 199 iterations.
4 total numbers solve in 200 iterations.
3 total numbers solve in 201 iterations.
21,623,484,475 of 25,922,280,429 total numbers did not solve out (83.42%). This percentage is increasing every time I finish a new set of numbers.
4,298,795,954 numbers have resolved into palindromes, so far.
PLEASE NOTE: These results are those ONLY from the numbers that my program checks. It DOES NOT check every single number. My algorithm determines which numbers can be eliminated from the search and still maintain 100% accurate results. As you can see from the stats above, my program actually only checked 25,922,280,429 numbers, not 9,999,999,999,999,999 numbers, which is a significant optimization. It would have taken over 150,000 years to compute all these numbers if I did not optimize my program.
Take care,
Jason Doucette
12/31/02 From: Dennis Nelsom To: Wade
Genesis 1:1 in Hebrew has 7 words and 28 letters with the Hebrew sum of 2701. Now 7x28 = 196. To me this number is easy to understand.
9/27/02 From: Jason Doucette To:Wade
Results of all 15-digit numbers:
60,000,252 total numbers solve in 1 iterations.
243,758,104 total numbers solve in 2 iterations.
162,203,350 total numbers solve in 3 iterations.
173,562,961 total numbers solve in 4 iterations.
132,415,634 total numbers solve in 5 iterations.
113,518,281 total numbers solve in 6 iterations.
99,243,179 total numbers solve in 7 iterations.
92,191,973 total numbers solve in 8 iterations.
82,044,294 total numbers solve in 9 iterations.
69,313,639 total numbers solve in 10 iterations.
67,559,522 total numbers solve in 11 iterations.
58,359,625 total numbers solve in 12 iterations.
52,992,006 total numbers solve in 13 iterations.
46,681,517 total numbers solve in 14 iterations.
44,838,265 total numbers solve in 15 iterations.
39,291,912 total numbers solve in 16 iterations.
35,976,236 total numbers solve in 17 iterations.
31,445,654 total numbers solve in 18 iterations.
28,955,797 total numbers solve in 19 iterations.
26,569,039 total numbers solve in 20 iterations.
23,683,472 total numbers solve in 21 iterations.
21,930,862 total numbers solve in 22 iterations.
19,391,621 total numbers solve in 23 iterations.
17,961,766 total numbers solve in 24 iterations.
16,170,388 total numbers solve in 25 iterations.
14,745,139 total numbers solve in 26 iterations.
13,270,723 total numbers solve in 27 iterations.
12,101,458 total numbers solve in 28 iterations.
10,916,581 total numbers solve in 29 iterations.
9,973,503 total numbers solve in 30 iterations.
9,027,740 total numbers solve in 31 iterations.
8,159,794 total numbers solve in 32 iterations.
7,401,295 total numbers solve in 33 iterations.
6,758,540 total numbers solve in 34 iterations.
6,110,693 total numbers solve in 35 iterations.
5,588,403 total numbers solve in 36 iterations.
5,104,389 total numbers solve in 37 iterations.
4,618,459 total numbers solve in 38 iterations.
4,198,230 total numbers solve in 39 iterations.
3,811,257 total numbers solve in 40 iterations.
3,466,236 total numbers solve in 41 iterations.
3,180,471 total numbers solve in 42 iterations.
2,884,269 total numbers solve in 43 iterations.
2,628,870 total numbers solve in 44 iterations.
2,383,297 total numbers solve in 45 iterations.
2,157,245 total numbers solve in 46 iterations.
1,971,414 total numbers solve in 47 iterations.
1,830,832 total numbers solve in 48 iterations.
1,675,596 total numbers solve in 49 iterations.
1,522,705 total numbers solve in 50 iterations.
1,378,202 total numbers solve in 51 iterations.
1,240,731 total numbers solve in 52 iterations.
1,148,649 total numbers solve in 53 iterations.
1,047,866 total numbers solve in 54 iterations.
943,276 total numbers solve in 55 iterations.
870,409 total numbers solve in 56 iterations.
778,996 total numbers solve in 57 iterations.
718,278 total numbers solve in 58 iterations.
662,005 total numbers solve in 59 iterations.
601,707 total numbers solve in 60 iterations.
554,469 total numbers solve in 61 iterations.
502,631 total numbers solve in 62 iterations.
459,988 total numbers solve in 63 iterations.
418,619 total numbers solve in 64 iterations.
385,122 total numbers solve in 65 iterations.
353,059 total numbers solve in 66 iterations.
322,403 total numbers solve in 67 iterations.
293,196 total numbers solve in 68 iterations.
265,015 total numbers solve in 69 iterations.
234,830 total numbers solve in 70 iterations.
214,153 total numbers solve in 71 iterations.
196,052 total numbers solve in 72 iterations.
179,171 total numbers solve in 73 iterations.
165,596 total numbers solve in 74 iterations.
149,625 total numbers solve in 75 iterations.
136,233 total numbers solve in 76 iterations.
125,840 total numbers solve in 77 iterations.
114,842 total numbers solve in 78 iterations.
103,128 total numbers solve in 79 iterations.
96,864 total numbers solve in 80 iterations.
88,089 total numbers solve in 81 iterations.
79,775 total numbers solve in 82 iterations.
73,904 total numbers solve in 83 iterations.
64,288 total numbers solve in 84 iterations.
58,903 total numbers solve in 85 iterations.
54,694 total numbers solve in 86 iterations.
50,108 total numbers solve in 87 iterations.
46,413 total numbers solve in 88 iterations.
44,130 total numbers solve in 89 iterations.
39,361 total numbers solve in 90 iterations.
36,582 total numbers solve in 91 iterations.
31,443 total numbers solve in 92 iterations.
28,885 total numbers solve in 93 iterations.
27,245 total numbers solve in 94 iterations.
25,455 total numbers solve in 95 iterations.
22,494 total numbers solve in 96 iterations.
21,067 total numbers solve in 97 iterations.
18,649 total numbers solve in 98 iterations.
16,911 total numbers solve in 99 iterations.
14,836 total numbers solve in 100 iterations.
14,605 total numbers solve in 101 iterations.
13,671 total numbers solve in 102 iterations.
12,109 total numbers solve in 103 iterations.
10,638 total numbers solve in 104 iterations.
9,979 total numbers solve in 105 iterations.
9,913 total numbers solve in 106 iterations.
8,899 total numbers solve in 107 iterations.
7,786 total numbers solve in 108 iterations.
6,665 total numbers solve in 109 iterations.
6,831 total numbers solve in 110 iterations.
5,858 total numbers solve in 111 iterations.
5,268 total numbers solve in 112 iterations.
4,832 total numbers solve in 113 iterations.
5,099 total numbers solve in 114 iterations.
3,861 total numbers solve in 115 iterations.
3,913 total numbers solve in 116 iterations.
3,604 total numbers solve in 117 iterations.
3,830 total numbers solve in 118 iterations.
3,120 total numbers solve in 119 iterations.
2,832 total numbers solve in 120 iterations.
2,750 total numbers solve in 121 iterations.
2,505 total numbers solve in 122 iterations.
2,446 total numbers solve in 123 iterations.
1,986 total numbers solve in 124 iterations.
1,839 total numbers solve in 125 iterations.
1,822 total numbers solve in 126 iterations.
1,288 total numbers solve in 127 iterations.
1,145 total numbers solve in 128 iterations.
1,035 total numbers solve in 129 iterations.
650 total numbers solve in 130 iterations.
627 total numbers solve in 131 iterations.
566 total numbers solve in 132 iterations.
505 total numbers solve in 133 iterations.
465 total numbers solve in 134 iterations.
504 total numbers solve in 135 iterations.
601 total numbers solve in 136 iterations.
553 total numbers solve in 137 iterations.
555 total numbers solve in 138 iterations.
606 total numbers solve in 139 iterations.
646 total numbers solve in 140 iterations.
401 total numbers solve in 141 iterations.
279 total numbers solve in 142 iterations.
382 total numbers solve in 143 iterations.
197 total numbers solve in 144 iterations.
148 total numbers solve in 145 iterations.
81 total numbers solve in 146 iterations.
22 total numbers solve in 147 iterations.
23 total numbers solve in 148 iterations.
73 total numbers solve in 149 iterations.
91 total numbers solve in 150 iterations.
46 total numbers solve in 151 iterations.
26 total numbers solve in 152 iterations.
8 total numbers solve in 153 iterations.
2 total numbers solve in 154 iterations.
11 total numbers solve in 155 iterations.
15 total numbers solve in 156 iterations.
5 total numbers solve in 157 iterations.
3 total numbers solve in 158 iterations.
1 total numbers solve in 159 iterations.
0 total numbers solve in 160 iterations.
0 total numbers solve in 161 iterations.
0 total numbers solve in 162 iterations.
137 total numbers solve in 163 iterations.
106 total numbers solve in 164 iterations.
33 total numbers solve in 165 iterations.
11 total numbers solve in 166 iterations.
3 total numbers solve in 167 iterations.
0 total numbers solve in 168 iterations.
40 total numbers solve in 169 iterations.
14 total numbers solve in 170 iterations.
52 total numbers solve in 171 iterations.
20 total numbers solve in 172 iterations.
11 total numbers solve in 173 iterations.
5 total numbers solve in 174 iterations.
0 total numbers solve in 175 iterations.
0 total numbers solve in 176 iterations.
0 total numbers solve in 177 iterations.
171 total numbers solve in 178 iterations.
89 total numbers solve in 179 iterations.
115 total numbers solve in 180 iterations.
97 total numbers solve in 181 iterations.
48 total numbers solve in 182 iterations.
123 total numbers solve in 183 iterations.
80 total numbers solve in 184 iterations.
100 total numbers solve in 185 iterations.
48 total numbers solve in 186 iterations.
22 total numbers solve in 187 iterations.
26 total numbers solve in 188 iterations.
16 total numbers solve in 189 iterations.
8 total numbers solve in 190 iterations.
5 total numbers solve in 191 iterations.
2 total numbers solve in 192 iterations.
0 total numbers solve in 193 iterations.
0 total numbers solve in 194 iterations.
0 total numbers solve in 195 iterations.
0 total numbers solve in 196 iterations.
0 total numbers solve in 197 iterations.
12 total numbers solve in 198 iterations.
7 total numbers solve in 199 iterations.
4 total numbers solve in 200 iterations.
3 total numbers solve in 201 iterations.
7,911,368,823 of 9,832,589,127 total numbers did not solve out (80.46%).
1,921,220,304 numbers have solved so far.
Again, please note:
These are only the numbers that I check - I do not check them all. My algorithm determines which numbers can be eliminated from the search which no consequences. As you can see from the stats above, I actually checked a total of 9,832,589,127 numbers, not 999,999,999,999,999 numbers. Quite a bit less, due to the optimizations. :)
Jason Doucette
8/19/02 From: Paul Leyland To: Wade
His Subject Line: 196 again, after all those years.
Hi,
I'm the PC Leyland you mention on your pages. The work you refer to was done about 20 years ago on a 4 MHz Z80-based machine running CP/M. The core reverse&add and the palindromicity detector were written in assembler and the I/O etc was written in Algol-60. The machine only had 32K of memory (actually rather a lot for those days) and I ran the program until it ran out of memory --- which explains the limit chosen for the number of iterations.
I must put a link to your page on my number theory curiosities page.
All the best,
Paul
8/19/02 From: Paul Leyland To: Wade
A crude page is now up at http://research.microsoft.com/users/cambridge/pleyland/CNT.htm under the Miscellanea and Curios link. The URL given is the one I prefer for public consumption as it is very much less likely to change than the substructure under it.
Other oddities will probably be added as my stock of round tuits is replenished.
Paul
8/19/02 From: Kirk Pearson To: Wade
You'll have a link from http://www.aspenleaf.com/distrib-upcoming.html later this evening or tomorrow. And when your project goes live, I'll move you to the ap-math.html page.
8/19/02 From: Corey Frang To: Wade
Your quest may be reaching "highly improbable", but you can test that theory by checking each iteration number for its "palindromic uncertainty", i.e. the number of digit pairs that sum to >9. If this number continuously grows, or if you were to graph the palindromic uncertainty of each iteration over about a million iterations, you would be able to get even more information on the possibility of finding the palindrome. If anything, I would be interested in seeing the graph :)
-Corey Frang
A Programmer for High-Voltage Software
www.high-voltage.com
8/18/02 From: Aaron Krowne To: Wade
Hi,
I came across your site today when following links from a slashdot story about Lychrel numbers. I thought it would be nice to have a PlanetMath entry about Lychrel numbers, so I put one up:
http://planetmath.org/encyclopedia/LychrelNumber.html
8/18/02 From: Matt Emmerton To: Wade
I read a lot of your site, and I have to say, the quest to determine palindromic numbers is quite fascinating!
The thing that struck me the most about some of the notes I read is that the number of carries in base-10 is crucial to determining how many iterations must be processed in order to achieve a palindromic number. (Indeed, this is the basis of the quotation in the "Probability" section on your site.)
Since I'm a computer scientist by trade (and a mathematician by schooling), I thought that I'd try out the problem in base-2 (binary). I find that many number problems relating to patterns provide much more obvious feedback when represented in base-2 as there are fewer states and situations that result in carries. (There are only 4 possible results from adding two digits (with carry-in) - 0, 0 with carry-out, 1 and 1 with carry-out).
Imagine my surprise:
196 decimal = 11000100 binary
11000100 + 00100011 = 11100111
Palindromic after one iteration!
What is clear from this is that since there are no carries, a palindromic result right away. This is already a known statement for base-10 (ie, 14 + 41 = 55), so I don't claim to be breaking any new ground.
What it does illustrate is the importance of carries in the quest to find palindromic numbers.
This has piqued my interest and I'm going to play around with doing stuff in different bases and see if that shows any patterns that aren't immediately obvious when working just in base 10.
196 is also a perfect square (14 * 14 = 196), so base 14 is a logical choice for a 1-iteration palindrome. I'll take a look at at the first bunch of perfect squares and see what characteristics are common amongst them as another avenue of investigation.
FYI, here's a quick chart of 196 represented in different bases, and the number of steps it takes to get to a palindromic number. (Sorry if the columns don't line up.)
NOTE: I have formatted his data into the following table:
| Base | Steps Until Palindrome | Total # of Carries | Progression |
2 3 4 4 6 7 8 9 10 11 12 13 14 15 16 17 18 |
1 2 3 4 4 1 1 2 Unsolved 1 0 0 1 1 2 2 3 |
0 4 2 4 7 0 0 2 Unsolved 0 0 0 0 0 2 1 3 |
11000100 + 00100011 = 1110 0111 21021 + 12012 = 100110 + 011001 = 111111 03010 + 01030 = 10100 + 00101 = 10201 1241 + 1421 = 3213 + 3123 = 10340 + 04301 = 20141 + 14102 = 34243 524 + 425 = 1353 + 3531 = 5324 + 4236 = 4003 + 3004 = 11011 400 + 004 = 404 304 + 403 = 707 237 + 732 = 1070 + 0701 = 1771 Unsolved 169 + 961 = A1A 141 (initial representation of 196 in base 12 is palindromic) 121 (initial representation of 196 in base 13 is palindromic) 100 + 001 = 101 D1 + 1D = EE C4 + 4C = 110 + 011 = 121 A9 + 9A = 132 + 231 = 363 AG + GA = 198 + 891 = A09 + 90A = 1111 |
--
Matt Emmerton
The following cluster of emails all came from the Slahdot Article on 8/18/02. I chose some of them for their points, and others for their support. I hope their authors do not mind my pasting their comments!!
by Anonymous Coward on Sunday August 18, @04:29PM (#4093873)
Albert Einstein failed out of high school and had no formal training in math....this guys brilliant!
While shifting numbers has (at first glance) has no fundamental significance rooted in formal methods, the story is quite intriguing considering how much effort people have put into carrying out pi. Maybe there is some odd parallel here....
by Anonymous Coward on Sunday August 18, @04:39PM (#4093911)
Not quite. There are all sorts of interesting phenomena that first get noticed experimentally. Fermat's Last Theorem and the Riemann-Zeta Hypothesis are two prime examples. The act of trying to prove conjectures about "neat" observations expands our understanding of the mathematics, even if we fail. To Bah-Humbug whole thing because there is no proof is to show a lack of wonder and amazement for the workings of the universe.
Bet you also liked to look at the back of the book for the answers instead of thinking about problems for yourself. ; )
by evilquaker on Sunday August 18, @09:21PM (#4094888)(User #35963 Info)
Re: "Until and unless there's a proof of why Lychrel numbers exist, the whole concept is quite uninteresting beyond a passing "neat"."
Actually, I think it's the opposite: when there's a proof either way, it will probably just be a mathematical curiosity (or it could turn out to be interesting, but I doubt it...). Until then, it's an unsolved problem. If you find the proof, you will most likely be the first person in the history of the Earth to know the answer. The fact that it's a relatively obscure problem, and that AFAIK no one has even come close to finding a method to attack the problem make it one of the better problems to work on if you like this feeling of adventure. Other problems (odd perfect number, Goldbach's, Twin Primes, Collatz) that fail one of the above two tests don't have this kind of promise.
by dragons_flight on Sunday August 18, @08:07PM (#4094653)(User #515217 Info | http://bounce.to/Bobby)
Given a 2*n digit number, it suffices to generate a palindrome if the sum of the i-th and (2n-i+1)-th digit is less than 10 for all i between 1 and n. It follows that at least (2n)/(2^n) numbers of length 2*n will immediately form palindromes. While less obvious, it is also true that if the sum of the i-th and (2n-i+1)-th digit is greater than 10 then the next iteration can only generate a palindrome if the sum of every digit and it's counterpart is greater than 10 (e.g. 9292 -> 12221). Not all numbers with this property will immediately form palindromes (e.g. 9393 -> 13332), but it is a requirement. This property holds for an additional (2n)/(2^n) numbers.
Hence the probability that a number of length 2*n will immediately form a palindrome is 1/(2^(n-1)) for each iteration.
On average, the number gains 0.5 digits per iteration of the algorithm. Consequently for a number with 2*n digits, after infinite iterations, you expect to have encountered a number of palindromes approximately equal to Sum(1/2^(n-1+k/2)), k=0 to infinity => ~6.8*2^(-n).
A straight forward density argument shows that there have to be some Lychrel numbers and that most numbers with a large number of digits are Lychrel numbers, but of course it doesn't tell you which particular numbers have this property.
Obviously I haven't been entirely rigorous, but after all this is slashdot.
by Uruk (mda at idatar.com) on Sunday August 18, @03:16PM (#4093521)(User #4907 Info | http://www.oldhat.org/freenet/index.html | Last Journal: Friday July 26, @04:48PM)
Since when does pure mathematics need to have an obvious application? Some people study math just because it's interesting. Sometimes, people come up with areas of number theory that don't immediately look promising, but that later get developed into something very useful, like optimum golomb rulers, or the mathematics that goes into public key crypto.
To get into the mind of a mathematician, you must understand the cardinal rule of math - that there is no such thing as an uninteresting number. All numbers have interesting aspects about them (strange prime factorizations, that they're palindromes, that they're the smallest sum of three consecutive cubes, whatever) but here's the real kicker - there's no such thing as an uninteresting number because if anybody was to ever find an uninteresting number that had absolutely nothing special about it, it would be interesting purely for the reason that it doesn't have anything interesting about that.,br>
Grasp that, and you can grasp why people do things like this. It's an intellectual exercise that some happen to like quite a bit.
by Anonymous Coward on Sunday August 18, @03:03PM (#4093469)
"Why do this?" Why calculate pi past the 10th digit?
Mathematics has fascinated man since the beginning of time (literally). There doesn't need to be any reason. I, for one, support this mans efforts.
Long live mathematic exploration!
by Archie Binnie (bastardspam.scottish@tightbastard.com) on Sunday August 18, @05:37PM (#4094115)(User #174447 Info | http://www.binnie.pwp.blueyonder.co.uk/)
Okay, so it's late and I was tired... but the first thing I tried when I reached for a calculator was reversing 196 (=691) and adding the sum of the digits (1+9+6). Adding them both together gives 707 (a palindrome).
Spooky or what?
But seriously, could this have anything to do with why it may have this odd property?
by ghastard (ryan03&visi,com) on Sunday August 18, @06:21PM (#4094257)(User #460282 Info)
Re: "Okay, so it's late and I was tired... but the first thing I tried when I reached for a calculator was reversing 196 (=691) and adding the sum of the digits (1+9+6). Adding them both together gives 707 (a palindrome)." Take this even further. By taking 196, and adding 16 (1+9+6) to it you get 212, another palindrome!
This is pretty cool.
by Anonymous Coward on Sunday August 18, @10:35PM (#4095091)
We've only got 10 digits (0123456789) and infinite places to put them.
it may be a really really really large number, but it will happen sooner or later.
I seriously doubt that
a) it can't happen eventually
b) anyone can prove that it can't happen eventually
in fact, if you took the 10 digits infinite slots, and algorithm approach you could probably prove that it will happen, but you'd have to prove that it will happen in X number of iterations.
by Anonymous Coward on Sunday August 18, @10:28PM (#4095076)
Seems to me any proof would start here.
What does it mean mathematically to reverse a number?
in some cases you make the new number larger (24 becomes 42)
some cases you make it smaller (42 becomes 24)
some cases nothing happens (44 becomes 44).... THESE cases are palindromes themselves.
interestingly enough, adding two palindromes doesn't always make another palindrome 44+44=88. 101+101=202 7337+7337=14674 but the algorithm stops here anyway.
You can easily classify those which get larger upon reversal by checking to see if the last digit is larger than the first. Same for those that get smaller.
Then some numbers have an even amount of digits, and some odd. Seems to me there is a balance property involved here. are even numbers more balanced, or odd? I guess odd are, because they'll share the middle digit (a pivot digit).
When adding the reverse, if the number gets smaller upon reversal, the resulting sum is closer to it's own value...not sure if that is useful, but I wonder if it's repeated.
i.e. if I take 804 and reverse it I get 408, then I add and I get 1212, closer to 804 than it is to 408. On the other hand, if I start with 408, and reverse it I get 804, and then add I get 1212, which is farther from the number I started with...which really means that all numbers only have to be done for digits ending in values less than or equal to 5.
For example you can solve 400,401,402,403,404,405...and stop...Then when you get into the 800s, solve 800,801,802,803,804,and 805. by solving 804 you solve 408..This can cut down the work. I suppose you can vice-versa this idea, but it's probably better to limit on the ones place than to try to limit the actual size of the number.
OK, so that should shave some time off of our calculating, what about the pivot number. In this case the number was 0, so that was shared, which means reversing doesn't introduce any change for that place. What if its shared but not zero.
415..514..929 (here the 2 is shared)..oh, wait. we got our palindrome...
wait, maybe it's not 5 in the last place, but rather if the last digit is larger than the current first digit that you skip. or vice versa
anyway, this way you can calculate whether to skip or not, instead of trying to keep track of those you've already found...691=196 in this respect.
weird, if you think about digits 0-1000, 555 is the absolute center of the universe.
END SLASHDOT POSTINGS.......
7/13/02 From: Jason To: Wade
The email is actually dated 7/6/02, It has just taken me this long to DO anything other than read about it. :-(
OK, here are the results after all 14-digit numbers were done:
15,000,188 total numbers solve in 1 iterations.43,464,944 total numbers solve in 2 iterations.
28,455,358 total numbers solve in 3 iterations.
31,279,587 total numbers solve in 4 iterations.
24,447,887 total numbers solve in 5 iterations.
21,197,531 total numbers solve in 6 iterations.
18,216,538 total numbers solve in 7 iterations.
16,708,547 total numbers solve in 8 iterations.
14,914,025 total numbers solve in 9 iterations.
12,509,875 total numbers solve in 10 iterations.
12,051,573 total numbers solve in 11 iterations.
10,362,411 total numbers solve in 12 iterations.
9,336,468 total numbers solve in 13 iterations.
8,222,360 total numbers solve in 14 iterations.
7,863,696 total numbers solve in 15 iterations.
6,863,280 total numbers solve in 16 iterations.
6,309,278 total numbers solve in 17 iterations.
5,533,096 total numbers solve in 18 iterations.
5,076,306 total numbers solve in 19 iterations.
4,602,863 total numbers solve in 20 iterations.
4,087,621 total numbers solve in 21 iterations.
3,785,924 total numbers solve in 22 iterations.
3,349,846 total numbers solve in 23 iterations.
3,093,771 total numbers solve in 24 iterations.
2,774,210 total numbers solve in 25 iterations.
2,524,552 total numbers solve in 26 iterations.
2,278,227 total numbers solve in 27 iterations.
2,078,372 total numbers solve in 28 iterations.
1,874,260 total numbers solve in 29 iterations.
1,712,866 total numbers solve in 30 iterations.
1,550,946 total numbers solve in 31 iterations.
1,403,076 total numbers solve in 32 iterations.
1,266,530 total numbers solve in 33 iterations.
1,163,632 total numbers solve in 34 iterations.
1,048,180 total numbers solve in 35 iterations.
956,507 total numbers solve in 36 iterations.
873,643 total numbers solve in 37 iterations.
789,316 total numbers solve in 38 iterations.
715,300 total numbers solve in 39 iterations.
649,522 total numbers solve in 40 iterations.
591,106 total numbers solve in 41 iterations.
543,574 total numbers solve in 42 iterations.
489,487 total numbers solve in 43 iterations.
450,256 total numbers solve in 44 iterations.
406,046 total numbers solve in 45 iterations.
368,591 total numbers solve in 46 iterations.
333,118 total numbers solve in 47 iterations.
306,220 total numbers solve in 48 iterations.
276,637 total numbers solve in 49 iterations.
252,243 total numbers solve in 50 iterations.
233,485 total numbers solve in 51 iterations.
212,194 total numbers solve in 52 iterations.
195,434 total numbers solve in 53 iterations.
178,012 total numbers solve in 54 iterations.
159,641 total numbers solve in 55 iterations.
150,871 total numbers solve in 56 iterations.
134,691 total numbers solve in 57 iterations.
126,179 total numbers solve in 58 iterations.
118,126 total numbers solve in 59 iterations.
105,942 total numbers solve in 60 iterations.
97,201 total numbers solve in 61 iterations.
87,251 total numbers solve in 62 iterations.
81,275 total numbers solve in 63 iterations.
73,684 total numbers solve in 64 iterations.
68,791 total numbers solve in 65 iterations.
62,136 total numbers solve in 66 iterations.
54,740 total numbers solve in 67 iterations.
50,788 total numbers solve in 68 iterations.
47,630 total numbers solve in 69 iterations.
42,294 total numbers solve in 70 iterations.
36,452 total numbers solve in 71 iterations.
32,708 total numbers solve in 72 iterations.
27,606 total numbers solve in 73 iterations.
25,719 total numbers solve in 74 iterations.
23,258 total numbers solve in 75 iterations.
21,915 total numbers solve in 76 iterations.
20,522 total numbers solve in 77 iterations.
20,185 total numbers solve in 78 iterations.
18,579 total numbers solve in 79 iterations.
16,623 total numbers solve in 80 iterations.
14,478 total numbers solve in 81 iterations.
13,357 total numbers solve in 82 iterations.
13,748 total numbers solve in 83 iterations.
12,240 total numbers solve in 84 iterations.
10,636 total numbers solve in 85 iterations.
9,401 total numbers solve in 86 iterations.
7,745 total numbers solve in 87 iterations.
6,266 total numbers solve in 88 iterations.
6,185 total numbers solve in 89 iterations.
5,817 total numbers solve in 90 iterations.
5,174 total numbers solve in 91 iterations.
4,458 total numbers solve in 92 iterations.
4,609 total numbers solve in 93 iterations.
4,536 total numbers solve in 94 iterations.
4,814 total numbers solve in 95 iterations.
4,173 total numbers solve in 96 iterations.
3,443 total numbers solve in 97 iterations.
3,258 total numbers solve in 98 iterations.
2,760 total numbers solve in 99 iterations.
2,705 total numbers solve in 100 iterations.
2,360 total numbers solve in 101 iterations.
2,385 total numbers solve in 102 iterations.
2,441 total numbers solve in 103 iterations.
2,018 total numbers solve in 104 iterations.
1,910 total numbers solve in 105 iterations.
1,762 total numbers solve in 106 iterations.
1,891 total numbers solve in 107 iterations.
1,541 total numbers solve in 108 iterations.
1,224 total numbers solve in 109 iterations.
987 total numbers solve in 110 iterations.
1,144 total numbers solve in 111 iterations.
890 total numbers solve in 112 iterations.
854 total numbers solve in 113 iterations.
854 total numbers solve in 114 iterations.
666 total numbers solve in 115 iterations.
713 total numbers solve in 116 iterations.
588 total numbers solve in 117 iterations.
812 total numbers solve in 118 iterations.
484 total numbers solve in 119 iterations.
698 total numbers solve in 120 iterations.
600 total numbers solve in 121 iterations.
507 total numbers solve in 122 iterations.
384 total numbers solve in 123 iterations.
321 total numbers solve in 124 iterations.
372 total numbers solve in 125 iterations.
431 total numbers solve in 126 iterations.
389 total numbers solve in 127 iterations.
288 total numbers solve in 128 iterations.
192 total numbers solve in 129 iterations.
96 total numbers solve in 130 iterations.
51 total numbers solve in 131 iterations.
98 total numbers solve in 132 iterations.
105 total numbers solve in 133 iterations.
83 total numbers solve in 134 iterations.
54 total numbers solve in 135 iterations.
41 total numbers solve in 136 iterations.
25 total numbers solve in 137 iterations.
11 total numbers solve in 138 iterations.
84 total numbers solve in 139 iterations.
61 total numbers solve in 140 iterations.
89 total numbers solve in 141 iterations.
81 total numbers solve in 142 iterations.
173 total numbers solve in 143 iterations.
94 total numbers solve in 144 iterations.
63 total numbers solve in 145 iterations.
38 total numbers solve in 146 iterations.
9 total numbers solve in 147 iterations.
4 total numbers solve in 148 iterations.
3 total numbers solve in 149 iterations.
0 total numbers solve in 150 iterations.
0 total numbers solve in 151 iterations.
0 total numbers solve in 152 iterations.
0 total numbers solve in 153 iterations.
0 total numbers solve in 154 iterations.
0 total numbers solve in 155 iterations.
0 total numbers solve in 156 iterations.
0 total numbers solve in 157 iterations.
0 total numbers solve in 158 iterations.
0 total numbers solve in 159 iterations.
0 total numbers solve in 160 iterations.
0 total numbers solve in 161 iterations.
0 total numbers solve in 162 iterations.
0 total numbers solve in 163 iterations.
0 total numbers solve in 164 iterations.
0 total numbers solve in 165 iterations.
0 total numbers solve in 166 iterations.
0 total numbers solve in 167 iterations.
0 total numbers solve in 168 iterations.
0 total numbers solve in 169 iterations.
0 total numbers solve in 170 iterations.
0 total numbers solve in 171 iterations.
0 total numbers solve in 172 iterations.
0 total numbers solve in 173 iterations.
0 total numbers solve in 174 iterations.
0 total numbers solve in 175 iterations.
0 total numbers solve in 176 iterations.
0 total numbers solve in 177 iterations.
0 total numbers solve in 178 iterations.
0 total numbers solve in 179 iterations.
0 total numbers solve in 180 iterations.
40 total numbers solve in 181 iterations.
18 total numbers solve in 182 iterations.
111 total numbers solve in 183 iterations.
78 total numbers solve in 184 iterations.
100 total numbers solve in 185 iterations.
48 total numbers solve in 186 iterations.
22 total numbers solve in 187 iterations.
10 total numbers solve in 188 iterations.
Note that these are only the numbers that I check - I do not check them all! My algorithm determines which numbers can be eliminated from the search which no consequences. So, when I say "10 total numbers solve in 188 iterations", it means 10 of the numbers I looked at, not 10 of all numbers of 14-digits or less. But, this should be fairly accurate in terms of a percentage.
Oh, one more thing:
1,017,226,026 of 1,364,330,547 total numbers did not solve out (74.56%). 347,104,521 numbers have solved so far.
There you can see how many numbers I actually checked- it is not 99,999,999,999,999!
Also, note that for all numbers under 10,000 (4 digits or less), 80% of them solve in 4 or less, and 90% of them solve in 7 or less. This is not the case as numbers get large.
6/30/02 From: Jason To: Wade
This was about the clustering of Lychrel Numbers around the powers of ten, that was on my Blackboard on 6/28/02.
Wade,
I believe the reason for this apparent 'cluster' (as you have already mentioned on your web page) is a result of the fact that you are only graphing the first number of each thread (thread meaning all numbers that converge into the same sequence). I have noticed this pattern a LONG time ago with my 'Most Delayed Palindromic Number' records: http://www.jasondoucette.com/worldrecords.html
You will notice almost ALL of the numbers start with 1. Why is this? Personally, I believe it is because of the reason I mentioned above. Remember, my program, as a huge optimization, starts with the first iteration of all numbers, and if I find something interested about that first iteration (such as a new world record!), then I have to find the number that creates that first iteration. In doing so, you notice that there are a ton of numbers that are possible, but most of the time (at least 50%), it is a number that starts with 1. Why? And does this have anything to do with Lychrel numbers?
Yes, it does, since, although Ben's program doesn't use the same specific optimizations as my program (i.e. he doesn't start with the 1st iteration numbers, and find the original starting number from this), his program produces the same results - he finds the smallest number that could have possibly created that first iteration (as all others will be deemed as copies in the sequence, and ignored).
To answer 'why': Since such a graph is only concerned with the first digit of the number, because you cannot tell 1,100,000 from 1,200,000 (well, you can a little, but hardly), and you certainly cannot see the difference from 1,000,100 and 1,000,200. So, let's look at the first digit of the number. When you have a 1st iteration in which you know the outer digits (first and last digit) must add to a specific number X, where X = 1 (0+1) through 18 (9+9), then there are many combinations of digits (0..9) that can produce that. The combinations that produce the smallest overall number always have a 1 as the first digit with a 0..9 as the last OR, a 9 as the last digit, and a 1..9 as the first. It just so happens that about 50% of the time the first case happens (at random), and about 50% of the time the second case happens. And this is with total random numbers - you get about 50% of them starting with 1.
If there exists a pattern outside of this, then, perhaps, you will get more than 50%, on average, that starts with 1. But if you get only 50%, then you are just getting whatever would happen randomly - thus concluding that there are no patterns.
Jason Doucette
6/30/02 From: Ben To: Wade
Hey there...
I've attached a .gif file which, if it makes any sense, will really blow you away (I packed a lot of info onto it which looks clear to me, but, I both know what it means and have rather bad color blindness). If the graph just looks like meaningless lines to you, let me know and I can give you a few smaller graphs that will make more sense (but take up more total screen space).
I calculated the delay for every number up to one billion, and made a log/log plot of it.
As the absolute most important feature, notice that delay in reaching a palindrome has fractal nature. This appears *very* obvious in the numbers that took only one or two steps to reach a palindrome (red or orange), and the resolution clearly increases with the range involved (notice the first three decades have fairly messy patterns, but by the fourth the fractal shows up nicely).
As the next most important feature, the exponent on the numbers that never reach a palindrome (white) noticeably exceeds that of the numbers that do reach a palindrome. This agrees with my earlier observation, using your table of Lychrels broken down by log magnitude, that the cardinality of the Lychrels seems to increase slightly less than twice as fast as that of all integers up to the same magnitude. Obviously, that trend cannot last indefinitely, but by extrapolating the trend, the number of Lychrels would (impossibly) exceed the number of integers before reaching one octillion (1E27). Although I cannot see any way to test this hypothesis on modern computing equipment, something "interesting" MUST happen well before reaching that point.
Finally, notice that the higher iteration count lines have a slightly higher exponent (apparent as slope on a log/log graph) than lower ones. This confirms your theory that the mean number of iterations required will grow with increasing starting value, though I would have to say it does not seem strong enough to explain the clustering of Lychrels from my previous graph.
Consider this wild speculation, but examining the 196 problem in terms of its fractal nature may well lead to an "explanation" of whether or not 196 will ever terminate, and why, based on Lychrels possibly lying at some sort of
critical points in the attractor, which will only resolve as we increase the resolution (although that might require
increasing it to infinity
- Ben
Here is Ben's graph:
6/29/02 From: Ben To: Wade
>> I don't know if you looked at the number of Lychrels in
>> a given number range, but it looks like this:
Hmm, no, I hadn't actually broken it down like that and considered it. Comparing against the hypothesis of Lychrels
having a "random" distribution, we would expect to see nine times as many for N+1 digits as for N digits (since nine
times as many numbers exist in that larger range). If anything, I would expect to see *less* than nine times as
many, since at least *some* should converge with earlier series. Yet, we seem to have somewhere around 17 times
as many, slightly less than 9*2 times more per order of magnitude. Obviously that trend cannot last, since at
some point the number of Lychrels would overtake the number of integers (a fairly rigid upper limit
>>I don't know how many iterations you have taken these to,
>>but it seems to me that the larger the starting number is,
>>the longer it should take to form a palindrome.
First, I need to explain that I've used the AI technique of "iterative deepening", where I consider "number of digits" as the measure of depth. So, because of that, I can more quickly check a set of known possible Lychrels to a greater depth (For example, the ones I sent you I ran to 500 and took two weeks, but checked to 5000 overnight). I first check them to 11 digits (which weeds out numbers that very quickly reach a palindrome), then to 40 digits (which eliminates all but a very small number of stragglers), then to 500 digits. Between 40 and 500, fewer than 1% dropped out as Lychrels. Between 500 and 5000, not a single one did (and, as I mentioned for the 1E8 data, I ran those *much* further and not a single one dropped out from 500 to whatever-i-ran-them-to (100,000 digits?)).
So, this seems to imply rather strongly that the numbers I sent you, while not "proven" as Lychrels, will not
fail as Lychrels for any digit length limit that we can reasonably test. I fully expect that *some* will
eventually prove non-Lychrels, but a handful at most (assuming, of course, that Lychrels exist at all... It
would certainly make us stop and think for a minute if 196 suddenly reached a palindrome, eh?
6/28/02 From: Wade To: Ben
Hey Ben....
I was thinking about this, and wanted your thoughts.....
I don't know if you looked at the number of Lychrels in a given number range, but it looks like this:
0 - 100 = 0
100 - 1,000 =2
1,000 - 10,000 =3
10,000 - 100,000 =69
100,000 - 1,000,000 =99
10,000,000 - 100,000,000 =1,728
100,000,000 - 1,000,000,000 = 29,813
A quick thought that I have had about the fact that there are so many more numbers for a higher range than a lower one was to wonder if they too will form out, but it will take a higher average number of iterations. (Is this the same as Jason's work?!?)
For example, if between 0 and 10,000, it takes an average of say, 5 iterations, to form a palindrome, and between 10,000 and 1,000,000 it takes an average of 60 iterations, (I'm just making up the numbers here) wouldn't it be logical that 100,000,000 to 1,000,000,000 would take say, 1,000 iterations or even 10,000,000?
For that matter, what *is* the average number of iterations that it takes to form a palindrome for the range 0-10,000 or 100,000,000 to 1,000,000,000? (There's ANOTHER program you can try to find time to write and I'll find a machine to run!!) :-)
I don't know how many iterations you have taken these to, but it seems to me that the larger the starting number is, the longer it should take to form a palindrome.
Either that, or as seems to be indicated on the chart and list that you provided, they will become more and more common, until eventually every number will be a Lychrel Number....
I don't know if you ever read that I changed the MSB of the 1 million data set, and ran it for 10 million iterations (ending archive entry 1/30/02), without a palindrome forming. I either hit a Lychrel Number by accident, or because they were more frequent, or because it required a far higher number of iterations to solve.
Any opinions?
6/12/02 Cut From: Ben To: Wade
First, regarding modulo checksums... The MOD-9 sum of digits will also equal the actual MOD-9 value of the number itself (trivially provable). Not very helpful, IMO, but perhaps someone can find a way to make use of it.
Also, an interesting pattern exists with the MOD-11 value (on the entire number, not the checksum)... After a few iterations of flip-and-add, it will always equal zero. I can't quite work out *why*, though it always follows the same pattern (identical MOD-11 values of the forward and reversed number, then on the last iteration before zero MOD-11, they will differ but add to 11... never taking more than 6 iterations).
Both of those result from similar properties under normal addition (casting out nines and casting out elevens), but behave somewhat different due to the symmetrical nature of how we manipulate the numbers. I currently wish to explore this property further, in the hopes of formalizing a more-or-less complete algebra under the given transformation (flipping and adding).