emirp

An emirp is a prime number that yields a different prime when its decimal digits are reversed. By definition, an emirp is not a palindrome, so reversing its digits does not reproduce the original number. If the reversed number is not prime or equals the original, the original number is not an emirp. The term emirp is a portmanteau of "prime" reversed.

Examples of emirp numbers include 13 (reversed 31), 17 (71), 37 (73), and 79 (97). More examples are 107 (701), 113 (311), 149 (941), 157 (751), 199 (991), and 311 (113). Pairs of emirps such as 13 and 31, or 37 and 73, are sometimes called emirp pairs.

Emirps must end with 1, 3, 7, or 9, since ending with any other digit would make the number composite or non-prime. Likewise, the reversed number cannot begin with 0; thus no emirp ends with 0, and zero cannot appear as the first digit. The concept is defined for base 10; analogues exist in other bases. It is not known whether there are infinitely many emirps.

The first few emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113. The set of emirps is catalogued in the OEIS as sequence A006567. Emirps are primarily studied in recreational and number-theory contexts as examples of primes closed under digit reversal.