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Lucky Palindromes: when do prime factors of palindromes make palindromes?

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POSTED BY: Vitaliy Kaurov
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POSTED BY: EDITORIAL BOARD

There is one construction method that leads to the majority of lucky palindromes so far. Take a prime whose digits are all ones or zeroes, with at most 9 ones, such that its digit reversal is also prime. Their product will not generate any carries, and is a (doubly lucky) palindrome, such as

121121010242121121383121121242010121121 == 11001000011000001011*11010000011000010011

There can also be a two and up to five ones, that just fits (note the central 9):

10201112123150905132121110201 == 100001001010201*102010100100001

And then there are the repunit primes. Combined with 11 they form palindromes of the form 122...221.

12222222222222222221 == 1111111111111111111*11 == 11*1111111111111111111

This happens to be the only lucky palindrome of length 20. I'll wrap up my search and report back with a followup post.
POSTED BY: Roman Maeder

Why do you exclude palindromes with a single (multiple) prime factor? they make for some pretty patterns, and in particular, a lucky palindrome that is a square has a prime factor that is itself a palindrome, like this 17 digit example:

12323244744232321 == 111010111 111010111

which is also a double lucky palindrome, of course. There are higher powers, as well:

343 == 7 7 7
14641 == 11 11 11 11
POSTED BY: Roman Maeder
POSTED BY: Vitaliy Kaurov

...with a single (multiple) prime factor

I see what you mean, those a interesting too. Do you see how frequent they are? Do you have an impression they are sensibly more frequent than cases with multiple distinct factors ?
POSTED BY: Vitaliy Kaurov

There are not that many. There are just two examples with 17 digits,

10022212521222001 == 100111001 100111001
12323244744232321 == 111010111 111010111

none with 18 digits (there's just a single, very lucky, palindrome with 18 digits) and none so far with 19 digits (search still ongoing):
POSTED BY: Roman Maeder
POSTED BY: Vitaliy Kaurov

The search can be optimized, and I have been running one for the last two hours or so. There are no lucky palindromes with 16 digits. There are a couple with 17 digits, and I just found one with 18 digits:

294507705507705492 == 2231118981118981 11 3 2 2

but no more than five factors so far.
POSTED BY: Roman Maeder

Wow, Roman, great, thank you! It would be really interesting to see the optimized code.

"...no lucky palindromes with 16 digits"

-- do you mean in both ascending and descending order?

294507705507705492 == 223111898111898111322

-- remarkable! And again 5 factors tops - very interesting. I wonder if 7-factors even possible.
POSTED BY: Vitaliy Kaurov

do you mean in both ascending and descending order?

both, I factor the number only once, and then test for both ascending and descending at the same time. They are just a Reverse[] apart.
POSTED BY: Roman Maeder
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