Palindromic prime

Palindromic prime
Conjectured no. of termsInfinite
First terms2, 3, 5, 7, 11, 101, 131, 151
Largest known term101888529 - 10944264 - 1
OEIS index
  • A002385
  • Palindromic primes: prime numbers whose decimal expansion is a palindrome

In mathematics, a palindromic prime (sometimes called a palprime[1]) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS)

Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. For any base, almost all palindromic numbers are composite,[2] i.e. the ratio between palindromic composites and all palindromes less than n tends to 1.

A large example,

101888529 - 10944264 - 1,

which has 1,888,529 digits, was found on 18 October 2021 by Ryan Propper and Serge Batalov.[3]

  1. ^ De Geest, Patrick. "World of Palindromic Primes". World!Of Numbers. Retrieved 1 April 2023.
  2. ^ Banks, William D.; Hart, Derrick N.; Sakata, Mayumi (2004). "Almost all palindromes are composite". Mathematical Research Letters. 11 (5–6): 853–868. arXiv:math/0405056. doi:10.4310/MRL.2004.v11.n6.a10. MR 2106245.
  3. ^ Chris Caldwell, The Top Twenty: Palindrome