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 decimal palindromic primes (sequence 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 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 palindromic primes include the Mersenne primes and the Fermat primes. The sequence of binary palindromic primes (, ) 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 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]
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