The following are some of the "statistical" things I've found about the individual data sets.
Table Legend
The "Iteration Count" is the first iteration that produces a given data set. If there were more than one iteration that produced say a 6 million digit number, I only saved the first one.
The formula I used for the "Iteration / Digit Ratio" is: I/D.
The formula I used for the "Digit Pairs Processed" is: Digits(Iterations+7)/2.
| Digit Milestone | Iteration Count | Iteration / Digit Ratio | Digit Pairs Processed |
1 Million 2 Million 3 Million 4 Million 5 Million 6 Million 7 Million 8 Million 9 Million 10 Million 11 Million 12 Million 13 Million 14 Million 15 Million 16 Million 17 Million 18 Million 19 Million 20 Million 21 Million 22 Million 23 Million 24 Million 25 Million 26 Million 27 Million 28 Million 29 Million 30 Million 31 Million 32 Million 33 Million 34 Million 35 Million 36 Million 37 Million 38 Million 39 Million 40 Million 41 Million 42 Million 43 Million 44 Million 45 Million 46 Million 47 Million 48 Million 49 Million 50 Million 51 Million 52 Million 53 Million 54 Million 55 Million 56 Million 57 Million 58 Million 59 Million 60 Million 61 Million 62 Million 63 Million 64 Million 65 Million 66 Million 67 Million 68 Million 69 Million 70 Million 71 Million 72 Million 73 Million 74 Million 75 Million 76 Million 77 Million 78 Million 79 Million 80 Million 81 Million 82 Million 83 Million 84 Million 85 Million 86 Million 87 Million 88 Million 89 Million 90 Million 91 Million 92 Million 93 Million 94 Million 95 Million 96 Million 97 Million 98 Million 99 Million 100 Million 101 Million 102 Million 103 Million 104 Million 105 Million 106 Million 107 Million 108 Million 109 Million 110 Million 111 Million 112 Million 113 Million 114 Million 115 Million 116 Million 117 Million 118 Million 119 Million 120 Million 121 Million 122 Million 123 Million 124 Million 125 Million 126 Million 127 Million 128 Million 129 Million 130 Million 131 Million 132 Million 133 Million 134 Million 135 Million 136 Million 137 Million 138 Million 139 Million 140 Million 141 Million 142 Million 143 Million 144 Million 145 Million 146 Million 147 Million 148 Million 149 Million 150 Million 151 Million 152 Million 153 Million 154 Million 155 Million 156 Million 157 Million 158 Million 159 Million 160 Million 161 Million 162 Million 163 Million 164 Million 165 Million 166 Million 167 Million 168 Million 169 Million 170 Million 171 Million 172 Million 173 Million 174 Million 175 Million 176 Million 177 Million 178 Million 179 Million 180 Million 181 Million 182 Million 183 Million 184 Million 185 Million 186 Million 187 Million 188 Million 189 Million 190 Million 191 Million 192 Million 193 Million 194 Million 195 Million 196 Million 197 Million 198 Million 199 Million 200 Million 201 Million 202 Million 203 Million 204 Million 205 Million 206 Million 207 Million 208 Million 209 Million 210 Million 211 Million 212 Million 213 Million 214 Million 215 Million 216 Million 217 Million 218 Million 219 Million 220 Million 221 Million 222 Million 223 Million 224 Million 225 Million 226 Million 227 Million 228 Million 229 Million 230 Million 231 Million 232 Million 233 Million 234 Million 235 Million |
2,415,836 4,830,292 7,245,928 9,662,071 12,078,865 14,495,294 16,911,863 19,327,667 21,744,278 24,159,531 26,574,642 28,991,159 31,408,295 33,824,775 36,240,602 38,655,227 41,070,365 43,487,182 45,901,512 48,316,988 50,732,545 53,148,015 55,565,809 57,982,436 60,398,130 62,816,046 65,231,462 67,648,377 70,067,270 72,482,579 74,899,358 77,314,955 79,731,034 82,145,364 84,563,227 86,979,674 89,397,247 91,812,894 94,228,899 96,645,213 99,061,141 101,476,346 103,893,138 106,309,010 108,723,539 111,138,308 113,554,695 115,969,543 118,387,016 120,802,305 123,218,044 125,634,154 128,050,160 130,463,975 132,880,223 135,295,252 137,712,060 140,128,242 142,544,153 144,958,406 147,375,122 149,790,895 152,207,835 154,623,289 157,038,002 159,455,103 161,871,411 164,287,044 166,704,687 169,121,766 171,538,507 173,953,453 176,368,856 178,784,747 181,199,797 183,616,094 186,032,643 188,447,228 190,862,368 193,277,613 195,693,825 198,109,015 200,524,477 202,939,323 205,355,672 207,771,197 210,186,316 212,602,940 215,018,872 217,433,686 219,850,418 222,266,315 224,680,750 227,095,084 229,512,704 231,928,518 234,344,371 236,759,404 239,175,318 241,590,277 244,006,839 246,423,282 248,840,359 251,254,880 253,670,404 256,084,270 258,498,530 260,914,435 263,328,945 265,745,475 268,160,697 270,575,863 272,991,794 275,405,890 277,820,730 280,236,256 282,651,618 285,066,604 287,481,429 289,896,907 292,311,641 294,727,521 297,143,533 299,557,657 301,974,027 304,389,137 306,805,299 309,221,256 311,637,141 314,052,527 316,468,413 318,884,030 321,299,093 323,715,600 326,131,076 328,547,998 330,963,887 333,379,370 335,795,486 338,210,775 340,626,948 343,043,659 345,459,815 347,875,673 350,292,487 352,708,290 355,125,177 357,540,909 359,956,876 362,372,813 364,788,891 367,204,161 369,619,640 372,035,743 374,450,389 376,866,208 379,281,305 381,697,126 384,113,355 386,529,610 388,944,754 391,360,944 393,776,305 396,192,233 398,607,648 401,023,403 403,439,651 405,856,956 408,272,435 410,688,792 413,104,065 415,517,439 417,933,403 420,349,873 422,765,244 425,181,055 427,599,027 430,016,536 432,433,414 434,849,911 437,267,293 439,682,132 442,097,346 444,512,168 446,927,453 449,343,164 449,343,164 454,174,655 456,588,593 459,004,983 461,419,218 463,835,090 466,250,388 468,667,675 471,085,391 473,503,339 475,919,151 478,335,425 480,750,860 483,166,859 485,581,911 487,996,904 490,413,021 492,829,843 495,245,922 497,662,056 500,077,852 502,495,484 504,913,664 507,329,950 509,745,400 512,162,233 514,577,796 516,992,665 519,408,582 521,825,691 524,242,175 526,658,409 529,075,350 531,491,865 533,907,350 536,322,341 538,737,263 541,153,446 543,569,184 545,986,095 548,402,519 550,818,415 553,233,019 555,650,203 558,065,511 560,480,471 562,896,706 565,311,914 In Work |
2 .4158360000 2 .4151460000 2 .4153093333 2 .4155177500 2 .4157730000 2 .4158823333 2 .4159804286 2 .4159583750 2 .4160308889 2 .4159531000 2 .4158765455 2 .4159299167 2 .4160226923 2 .4160553571 2 .4160401333 2 .4159516875 2 .4159038235 2 .4159545556 2 .4158690526 2 .4158494000 2 .4158354762 2 .4158188636 2 .4159047391 2 .4159348333 2 .4159252000 2 .4160017692 2. 4159800741 2. 4160134643 2. 4161127586 2. 4160859667 2. 4161083226 2. 4160923438 2. 4160919394 2. 4160401176 2. 4160922000 2. 4161020556 2. 4161418108 2. 4161287895 2. 4161256154 2. 4161303250 2. 4161253902 2. 4161034762 2. 4161194884 2. 4161138636 2. 4160786444 2. 4160501739 2. 4160573404 2. 4160321458 2. 4160615510 2. 4160461000 2. 4160400784 2. 4160414231 2. 4160407547 2. 4159995370 2. 4160040545 2. 4159866429 2. 4160010526 2. 4160041724 2. 4160025932 2. 4159734333 2. 4159856066 2. 4159821774 2. 4159973810 2. 4159888906 2. 4159692615 2. 4159864091 2. 4159912090 2. 4159859412 2. 4160099565 2. 4160252286 2. 4160353099 2. 4160201806 2. 4160117260 2. 4160100946 2. 4159972933 2. 4160012368 2. 4160083506 2. 4159901026 2. 4159793418 2. 4159701625 2. 4159731481 2. 4159635976 2. 4159575542 2. 4159443214 2. 4159490824 2. 4159441512 2. 4159346667 2. 4159425000 2. 4159423820 2. 4159298444 2. 4159386593 2. 4159382065 2. 4159220430 2. 4159051489 2. 4159232000 2. 4159220625 2. 4159213505 2. 4159122857 2. 4159123030 2. 4159027700 2. 4159092970 2. 4159145294 2. 4159258155 2. 4159123077 2. 4159086095 2. 4158893396 2. 4158741121 2. 4158743981 2. 4158618807 2. 4158679545 2. 4158621351 2. 4158559196 2. 4158565841 2. 4158411404 2. 4158324348 2. 4158297931 2. 4158257949 2. 4158186780 2. 4158103277 2. 4158075583 2. 4157986860 2. 4157993525 2. 4158010813 2. 4157875565 2. 4157922160 2. 4157868016 2. 4157897559 2. 4157910625 2. 4157917907 2. 4157886692 2. 4157894122 2. 4157881061 2. 4157826541 2. 4157880597 2. 4157857481 2. 4157941029 2. 4157947956 2. 4157925362 2. 4157948633 2. 4157912500 2. 4157939574 2. 4158004155 2. 4158029021 2. 4158032847 2. 4158102552 2. 4158102055 2. 4158175306 2. 4158169527 2. 4158179597 2. 4158187533 2. 4158204702 2. 4158168487 2. 4158146405 2. 4158165130 2. 4158089613 2. 4158090256 2. 4158044904 2. 4158045949 2. 4158072642 2. 4158100625 2. 4158059255 2. 4158082963 2. 4158055521 2. 4158062988 2. 4158039273 2. 4158036325 2. 4158062934 2. 4158152143 2. 4158132249 2. 4158164235 2. 4158132456 2. 4157990640 2. 4158000173 2. 4158038678 2. 4158013943 2. 4158014489 2. 4158137119 2. 4158232360 2. 4158291285 2. 4158328389 2. 4158413978 2. 4158358901 2. 4158324918 2. 4158270000 2. 4158240703 2. 4158234624 2. 4158237594 2. 4158226330 2. 4158126614 2. 4158157000 2. 4158074241 2. 4158077604 2. 4158051192 2. 4158127577 2. 4158225179 2. 4158333622 2. 4158332538 2. 4158354798 2. 4158334673 2. 4158342950 2. 4158304030 2. 4158262574 2. 4158276897 2. 4158325637 2. 4158337659 2. 4158352233 2. 4158350338 2. 4158436731 2. 4158548517 2. 4158569048 2. 4158549763 2. 4158595896 2. 4158581972 2. 4158535748 2. 4158538698 2. 4158596806 2. 4158625576 2. 4158642615 2. 4158691781 2. 4158721136 2. 4158703620 2. 4158664009 2. 4158621659 2. 4158635982 2. 4158630400 2. 4158676770 2. 4158701278 2. 4158702412 2. 4158647118 2. 4158704478 2. 4158680130 2. 4158640991 2. 4158656910 2. 4158628803 In Work |
1,207,921,500,000 4,830,299,000,000 10,868,902,500,000 19,324,156,000,000 30,197,180,000,000 43,485,903,000,000 59,191,545,000,000 77,310,696,000,000 97,849,282,500,000 120,797,690,000,000 146,160,569,500,000 173,946,996,000,000 204,153,963,000,000 236,773,474,000,000 271,804,567,500,000 309,241,872,000,000 349,098,162,000,000 391,384,701,000,000 436,064,430,500,000 483,169,950,000,000 532,691,796,000,000 584,628,242,000,000 639,006,884,000,000 695,789,316,000,000 754,976,712,500,000 816,608,689,000,000 880,624,831,500,000 947,077,376,000,000 1,015,975,516,500,000 1,087,238,790,000,000 1,160,940,157,500,000 1,237,039,392,000,000 1,315,562,176,500,000 1,396,471,307,000,000 1,479,856,595,000,000 1,565,634,258,000,000 1,653,849,199,000,000 1,744,445,119,000,000 1,837,463,667,000,000 1,932,904,400,000,000 2,030,753,534,000,000 2,131,003,413,000,000 2,233,702,617,500,000 2,338,798,374,000,000 2,446,279,785,000,000 2,556,181,245,000,000 2,668,535,497,000,000 2,783,269,200,000,000 2,900,482,063,500,000 3,020,057,800,000,000 3,142,060,300,500,000 3,266,488,186,000,000 3,393,329,425,500,000 3,522,527,514,000,000 3,654,206,325,000,000 3,788,267,252,000,000 3,924,793,909,500,000 4,063,719,221,000,000 4,205,052,720,000,000 4,348,752,390,000,000 4,494,941,434,500,000 4,643,517,962,000,000 4,794,547,023,000,000 4,947,945,472,000,000 5,103,735,292,500,000 5,262,018,630,000,000 5,422,692,503,000,000 5,585,759,734,000,000 5,751,311,943,000,000 5,919,262,055,000,000 6,089,617,247,000,000 6,262,324,560,000,000 6,437,463,499,500,000 6,615,035,898,000,000 6,794,992,650,000,000 6,977,411,838,000,000 7,162,257,025,000,000 7,349,442,165,000,000 7,539,063,812,500,000 7,731,104,800,000,000 7,925,600,196,000,000 8,122,469,902,000,000 8,321,766,086,000,000 8,523,451,860,000,000 8,727,616,357,500,000 8,934,161,772,000,000 9,143,105,050,500,000 9,354,529,668,000,000 9,568,340,115,500,000 9,784,516,185,000,000 10,003,194,337,500,000 10,224,250,812,000,000 10,447,655,200,500,000 10,673,469,277,000,000 10,901,853,772,500,000 11,132,569,200,000,000 11,365,702,333,000,000 11,601,211,139,000,000 11,839,178,587,500,000 12,079,514,200,000,000 12,322,345,723,000,000 12,567,587,739,000,000 12,815,278,849,000,000 13,065,254,124,000,000 13,317,696,577,500,000 13,572,466,681,000,000 13,829,671,729,500,000 14,089,379,868,000,000 14,351,427,884,000,000 14,616,001,510,000,000 14,882,919,072,000,000 15,152,248,720,000,000 15,424,036,756,500,000 15,698,136,129,000,000 15,974,692,377,500,000 16,253,703,254,000,000 16,535,120,062,500,000 16,818,930,049,000,000 17,105,145,442,000,000 17,393,814,840,000,000 17,684,854,704,000,000 17,978,379,208,000,000 18,274,327,710,000,000 18,572,575,168,000,000 18,873,377,125,000,000 19,176,516,072,000,000 19,482,136,931,000,000 19,790,160,832,000,000 20,100,596,046,000,000 20,413,414,710,000,000 20,728,681,510,000,000 21,046,346,442,000,000 21,366,390,150,000,000 21,688,945,669,000,000 22,013,848,102,500,000 22,341,264,340,000,000 22,671,026,739,000,000 23,003,177,013,000,000 23,337,786,763,500,000 23,674,754,740,000,000 24,014,200,327,500,000 24,356,100,286,000,000 24,700,377,273,000,000 25,047,048,960,000,000 25,396,205,815,000,000 25,747,705,681,000,000 26,101,701,024,000,000 26,458,027,784,000,000 26,816,787,783,500,000 27,177,961,500,000,000 27,541,561,799,000,000 27,907,516,768,000,000 28,275,902,995,500,000 28,646,752,750,000,000 29,019,905,690,000,000 29,395,564,770,000,000 29,773,582,992,000,000 30,154,073,507,000,000 30,537,012,279,000,000 30,922,369,360,000,000 31,310,053,260,500,000 31,700,237,031,000,000 32,092,769,428,000,000 32,487,763,680,000,000 32,885,131,537,500,000 33,284,943,030,000,000 33,687,211,443,000,000 34,091,984,892,000,000 34,499,021,349,000,000 34,908,547,915,000,000 35,320,398,156,000,000 35,734,500,356,000,000 36,151,239,965,000,000 36,570,439,560,000,000 36,991,959,462,500,000 37,415,933,456,000,000 37,842,514,509,000,000 38,271,472,327,000,000 38,702,791,179,500,000 39,136,492,620,000,000 39,572,690,650,000,000 40,011,074,649,000,000 40,451,907,799,500,000 40,895,120,100,000,000 41,340,790,050,000,000 41,788,914,903,000,000 42,239,471,175,000,000 42,692,418,228,000,000 43,147,622,700,000,000 43,605,474,050,000,000 44,065,535,987,500,000 44,528,169,312,000,000 44,993,163,117,500,000 45,460,765,154,000,000 45,930,826,305,000,000 46,403,327,908,000,000 46,878,037,063,000,000 47,355,207,768,000,000 47,834,711,266,500,000 48,316,686,600,000,000 48,800,982,759,000,000 49,287,688,011,000,000 49,776,922,342,000,000 50,268,644,700,000,000 50,762,707,722,500,000 51,259,192,489,000,000 51,758,058,406,500,000 52,259,531,064,000,000 52,763,478,619,500,000 53,269,645,485,000,000 53,778,140,438,500,000 54,289,197,440,000,000 54,802,536,019,500,000 55,318,215,904,000,000 55,836,423,317,500,000 56,357,175,384,000,000 56,880,276,747,000,000 57,405,767,344,000,000 57,933,751,591,500,000 58,464,105,920,000,000 58,996,762,948,500,000 59,531,780,628,000,000 60,069,205,605,000,000 60,609,186,736,000,000 61,151,533,987,500,000 61,696,429,526,000,000 62,243,686,701,000,000 62,793,300,108,000,000 63,345,181,477,000,000 63,899,774,150,000,000 64,456,567,329,000,000 65,015,735,448,000,000 65,577,467,064,500,000 66,141,494,757,000,000 In Work |
NOTE: Berend Jan van der Zwaag of the Netherlands has brought to my attention that the palindromes checked for in the table below are possibly in error. His results differ from my own. I am looking into it, and will correct them as I find out who is correct. (Although I suspect that it's probably Berend who is correct....)
Below is a some data of the palindromes found within some files of very large numbers. Most of the data files are the results from this 196 quest, but in order to have a larger sample, I gathered some number files off of the web. I do not care if the calculations of the number files are accurate, (except for the 196 files I DO WANT THOSE TO BE ACCURATE!! :-) ) my goal was to just gather some very large number sets, and test them for palindromes.
Table Legend
The "Data Set Basis" is a general idea of what was being calculated to find the number. Most of them are my own 196 files.
"File Length" is the number of digits in the file claimed by the author. I didn't count the digits in other people's files, so I just have to trust that they were careful.
"File Location" is where I found the file from on the internet. Again, most of these are my own local files, but I provide a URL to the ones I can, from the internet, in case someone would like to verify my findings. (This would not hurt my feelings!!!)
"# of Even Pals" is how many palindromes of 6 digits long or greater that I found, which have an EVEN number of digits.
"# of Odd Pals" is how many palindromes of 7 digits long or greater that I found, which have an ODD number of digits.
"Percentage" is the percentage of even finds to odds.
| Data Set Basis | File Length | File Location | # of Even Pals | # of Odd Pals | Percentage |
SQRT 2 SQRT 5 SQRT 6 SQRT 7 SQRT 8 SQRT 10 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc 196 calc |
2,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000 9,000,000 10,000,000 11,000,000 12,000,000 13,000,000 14,000,000 15,000,000 16,000,000 17,000,000 18,000,000 19,000,000 20,000,000 21,000,000 22,000,000 23,000,000 24,000,000 25,000,000 26,000,000 27,000,000 28,000,000 29,000,000 30,000,000 31,000,000 32,000,000 33,000,000 34,000,000 35,000,000 36,000,000 37,000,000 38,000,000 39,000,000 40,000,000 |
Robert Nemiroff and Jerry Bonnell Robert Nemiroff and Jerry Bonnell Robert Nemiroff and Jerry Bonnell Robert Nemiroff and Jerry Bonnell Robert Nemiroff and Jerry Bonnell Robert Nemiroff and Jerry Bonnell Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local file Local File Local file Local file Local file |
948 655 625 628 591 646 722 1,033 1,229 1,304 1,559 1,583 1,532 1,669 1,659 2,435 2,134 2,061 4,641 2,098 2,290 2,624 2,660 2,426 2,700 3,036 2,997 3,111 3,219 2,713 5,161 7,266 2,999 3,608 3,773 3,406 3,925 4,054 3,815 4,391 3,548 4,074 3,848 3,814 4,231 4,699 |
1,649 906 918 905 872 868 887 1,754 2,401 3,154 3,687 4,334 4,645 5,320 5,643 6,164 6,434 6,928 6,925 7,317 7,925 7,758 8,041 8,702 8,910 9,036 8,945 9,324 9,404 10,028 10,249 9,978 10,631 10,264 10,438 10,550 10,676 10,782 11,218 11,336 11,607 11,650 11,739 11,930 12,005 11,870 |
57 72 68 69 67 74 81 59 51 41 42 37 33 31 29 40 33 30 67 29 29 34 33 28 30 34 34 33 34 27 50 73 28 35 36 32 37 38 34 39 31 35 33 32 35 40 |
I find this very interesting!! There are the same number of even digit palindromes as odd ones, from here to infinity. Yet, in every single test case above, there are about 3 times as many odd digit palindromes as evens. Even though I am technically starting with a harder search, because there are more digits. (This holds true for the other data sets that I don't have listed above, as well...)
If I had to guess the reason for this, I would say that it has to do with the probability of the double digit in the middle of the evens.
Richard Farmbrough from the United Kingdom sent me the following:
I am interested in your observations on even/odd pal distributions.
Your suggested reason (double digit) is not valid because XX has the same probability as XYX (where Y can be = or <> X).In both cases this is 10%.
Rgds,
Richard.
I've never seen anyone discussing this on any of the pages that I have read. Does anyone have any input??