Palindromic primes

Palindromic primes sit at the junction of two very different traditions. One is the recreational numerology of digits, which has no theorems in it at all; the other is the arithmetic of divisibility, which has one sharp result to contribute. A number is divisible by eleven exactly when the alternating sum of its digits is. Reverse a palindrome of even length and the alternating sum cancels to zero, so every even-length palindrome — 1221, 45654554, all of them — is a multiple of eleven. Eleven itself is the single exception that is also prime. One line of elementary arithmetic therefore wipes out half of all candidates forever, which is unusually good value.

The sequence's low catalogue number, A002385, places it among the entries Neil Sloane gathered for A Handbook of Integer Sequences in 1973, the printed ancestor of the On-Line Encyclopedia of Integer Sequences. Martin Gardner's "Mathematical Games" column in Scientific American had already carried palindromic numbers to a far wider audience, usually in company with the reverse-and-add problem of whether 196 ever turns into a palindrome, which remains unsolved.

Beyond that the record is one of computation rather than proof. Harvey Dubner spent decades hunting large palindromic primes with increasingly capable hardware, and the current records run past a million digits — found, as large primes almost always are now, by distributed search rather than by insight. The one genuine theorem of the modern era points the wrong way for collectors: in 2004 William Banks, Derrick Hart and Mayumi Sakata proved that almost all palindromes are composite, meaning the primes among them have density zero. Whether infinitely many palindromic primes exist at all is still open.

Key properties

• Apart from 11, every palindromic prime has an odd number of digits: an even-length palindrome is always divisible by 11.
• There are exactly 15 three-digit palindromic primes, and exactly 20 palindromic primes below 10,000.
• A palindromic prime above 11 cannot start with 2, 4, 5, 6 or 8, because its first and last digits are equal and no prime above 5 ends in an even digit or in 5.
• The four single-digit primes 2, 3, 5 and 7 are all trivially palindromic.
• Repunits — 1, 11, 111, … — are palindromes, so every repunit prime is a palindromic prime; the smaller known ones have 2, 19, 23, 317 and 1031 digits.
• Being palindromic depends on the base: 7 is a palindrome in decimal and in binary (111), while 13 is the palindrome 11 in base 12 but 1101 in binary.
• In 2004 Banks, Hart and Sakata proved that almost all palindromes are composite, so palindromic primes have density zero among palindromes.
• Whether there are infinitely many palindromic primes is an open question.

Where they turn up

• The 31-digit prime 1000000000000066600000000000001 — 666 flanked by thirteen zeros on each side — was nicknamed "Belphegor's prime" by the writer Clifford Pickover; the name is whimsy, the primality is not.
• Repunit primes, which are a subfamily of palindromic primes, are tracked by the Cunningham Project and are standard test cases for factorisation software.
• Mersenne primes are palindromes in binary, since 2^p - 1 is a run of p ones — the same digit-reversal idea in a different base.
• Palindromes are a staple of introductory programming exercises and benchmarks; Project Euler's fourth problem asks for the largest palindromic product of two three-digit numbers.
• Palindromic numbers are treated as auspicious in several numerological traditions and in some date-watching customs — a belief about digits rather than a mathematical property.