Palindromic prime

A palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns. The first few decimal palindromic primes (sequence A002385 in OEIS) are:

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, …

It may be noticed that in the above list there are no 2- or 4-digit palindromic primes, except for 11. If one considers the divisibility test for 11, it can be deduced that any palindromic number with an even number of digits is divisible by 11. Therefore, except for 11, all palindromic primes have an odd number of digits.

It is not known if there are infinitely many palindromic primes in base 10. The largest known is 10180004 + 248797842 × 1089998 + 1, found by Harvey Dubner in 2007.

In binary, the palindromic primes include the Mersenne primes and the Fermat primes. The sequence of binary palindromic primes (A117697, A016041) begins:

binary: 11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, 10010101001, …

decimal: 3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193,

Ribenboim defines a triply palindromic prime as a prime p for which: p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime. For example, p = 1011310 + 4661664 × 105652 + 1, which has q = 11311 digits, and 11311 has r = 5 digits. The first (base-10) triply-palindromic prime is the 11-digit 10000500001. It's possible that a triply palindromic prime in base 10 may also be palindromic in another base, such as base 2, but it would be highly remarkable if it was also a triply palindromic prime in that base as well.

References

* Chris Caldwell, The Top Twenty: Palindrome

* Paulo Ribenboim, The New Book of Prime Number Records

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