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 OEIS|id=A002385 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 + 248797842e|89998 + 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 (OEIS2C|id=A117697, OEIS2C|id=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 + 4661664e|5652 + 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.


* Chris Caldwell, [ "The Top Twenty: Palindrome"]
* Paulo Ribenboim, "The New Book of Prime Number Records"

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • palindromic prime — noun A prime number that is also palindromic, its digits read backwards are the same as read forward. 17471 is a palindromic prime …   Wiktionary

  • Palindromic number — A palindromic number or numeral palindrome is a symmetrical number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under… …   Wikipedia

  • Palindromic sequence — A palindrome is a word that is the same forward as it is backward such as the word racecar , the name Otto , or the saying attributed to Napoleon Able was I ere I saw Elba . Certain sequences of DNA are the same whether they are read 5 (five… …   Wikipedia

  • Dihedral prime — A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven segment display, regardless of orientation (normally or upside down), and surface (actual display or… …   Wikipedia

  • Strobogrammatic prime — A strobogrammatic prime is a prime number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down. In base 10, given a set of glyphs where 0, 1 and 8 are symmetrical around the horizontal axis, and 6… …   Wikipedia

  • Strictly non-palindromic number — A strictly non palindromic number is an integer n that is not palindromic in any numeral system with a base b in the range 2 le; b le; n − 2. For example, the number six is written as 110 in base 2, 20 in base 3 and 12 in base 4, none of which is …   Wikipedia

  • List of prime numbers — This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. By Euclid s theorem, there are an infinite number of prime numbers. Subsets of the… …   Wikipedia

  • Mersenne prime — Named after Marin Mersenne Publication year 1536[1] Author of publication Regius, H. Number of known terms 47 Conjectured number of terms Infinite …   Wikipedia

  • Cuban prime — A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is: and the first few cuban primes from this equation are (sequence A002407 in OEIS): 7 …   Wikipedia

  • Super-prime — The subsequence of prime numbers that occupy prime numbered positions within the sequence of all prime numbers begins:3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, … OEIS|id=A006450.That is, if p ( i ) denotes the i th prime number, the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”