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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: 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

Great question. Because there is no asymmetry. It is easy to get a palindrome from 2 or more identical palindromes. But it is much harder for not-identical non-palindromic factors to line up to form a palindrome. I am just curious about these not-trivial cases, if that makes sense.
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

I actually agree, Roman, single multiple prime factor are also very nice. When you done with your search I'd love to see the sequences complete with those numbers and the optimized code. Thank you very much for looking into this.
POSTED BY: Vitaliy Kaurov
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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