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196 AND OTHER LYCHREL NUMBERS

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Welcome to p196.org!

Note from Vaughn Suite to Wade.

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!!!!!!!!!!!!!!

65
65
65
65

ad infinitum.

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 greater than 10), or it is the same as the last digit (the last sum was less than 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 greater than 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....

92116 iterations 38243 digits
126528 iterations 52494 digits

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.]

LL...92116 iterations 38243 digits
LL...126528 iterations 52494 digits
LL...211282 iterations 87571 digits
LL...231575 iterations 95920 digits
LL...237883 iterations 98535 digits

[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...