Palindromic prime
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A palindromic prime 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 palindromic primes in base 10 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, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, 18481, 19391, 19891, 19991
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 any palindromic number with an even number of digits is divisible by 11.
It is not known if there are infinitely many palindromic primes in base 10. The largest known palindromic prime is 10130022 + 3761673 × 1065008 + 1, which was found by Harvey Dubner on November 7 2004 and announced on November 19 2004.
In binary, the easiest palindromic primes to find are Mersenne primes, since they are also repunit primes. The first four non-Mersenne palindromic primes in binary are 5 (101), 17 (10001), 73 (1001001) and 107 (1101011).
Ribenboim defines a triply palindromic prime as one, which, in addition to being a palindromic prime, also has a number of digits which is itself a palindromic prime. For example, 1011310 + 4661664 x 105652 + 1, which has 11311 digits. It's possible that a triply palindromic prime in base 10 may be 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
- Paulo Ribenboim, The New Book of Prime Number Records
- Harvey Dubner, "New palindromic prime record", posted to the NMBRTHRY mailing list (http://listserv.nodak.edu/scripts/wa.exe?A2=ind0411&L=nmbrthry&F=&S=&P=518)
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