I read your papers. I noticed some similar points (as we say in Trinidad, GREAT MINDS THINK ALIKE!!! - the other part is "and fools seldom differ", but in this case it is obvious that you are not being foolish :-) )

I decided to modify my program to see when the sums of the first and the digits allow for a palindrome with a carry, in other words, whether they add up to 11. I printed the first and last digits for all those that added to 11. The results are striking!!!!!!!!!!!!!!

At first I wondered if I did something wrong. But then it makes sense. After repeated reverse and adds, the first digit is eather 1 (the last sum was > 10), or it is the same as the last digit (the last sum was <10 with no carry into that digit from the second to last and second digit), or it is 1 more than the last digit (the last sum was <10 with a carry from the sum of the second and second to last digit).

You should never find, after even 1 reverse and add, a first digit of 3 and a last digit of 8, for example. I hope this information helps in the general search for a non-brute force approach.

PS. In the short paper, paper4, you said, "In 127,986 iterations of the Lychrel number 196, there is only a SINGLE iteration of the pattern of "LL..." (11/11/x). It is "LLB..." (11/11/1...) and occurs 51,909 iterations into the set."

I stand to be corrected, but my program shows a pattern of "LL..." several twice before iteration 127,986, and 51,909 is not it....

I really cannot tell you what the other digits are right now... I'll mod the program and tell you tomorrow.

[Okay, I had overwritten my data for up to >1,000,000 iterations in testing, so I just reiterated the previous program to 251,528 iterations.]

[Plus I fixed the program to give the new data up to 214,218 iterations {that's when I interrupted it} ... I could not go any further since I am on a 550MHz k6-2, which runs pbcdchk about 30 times slower than your machine does]

The first three are LLA with the a being 0+1.

You should be able to use the programs I modified specially for the 1, 2, 3, 4 digit output to investigate this further. I'll probably let it run an hour or so when I get it to a Pentium 4...

Cheers,

Vaughn

Note from Wade I'm looking into the "error" Vaughn brings up... I'll corect the page and update this when I find it...