Emirp

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Class of prime numbers

An emirp (pronounced or , an anadrome of prime) is a prime number that results in a different prime when its decimal digits (digits in base 10) are reversed.^[1] This definition excludes the related palindromic primes. The term reversible prime is used to mean the same as emirp, but may also, ambiguously, include the palindromic primes.

The sequence of emirps begins 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, ... ‹See TfM›(sequence A006567 in the OEIS).^[1]

The difference in all pairs of emirps is always a multiple of 18. This follows from all primes bigger than 2 being odd (making their differences even, i.e. multiples of 2) and from differences between pairs of natural numbers with reversed digits being multiples of 9 (which itself is a consequence of ${\displaystyle 10^{n}-1}$ being a multiple of 9 for every non-negative integer ${\displaystyle n}$).

All non-palindromic permutable primes are emirps.

It is not known whether there are infinitely many emirps. The largest known emirp as of April 19, 2026 is 10¹¹¹⁹⁵⁶ - 7*10⁵³⁸⁵⁵ - 1 by Ryan Propper and Serge Batalov.^[2] The integer's reverse is 10¹¹¹⁹⁵⁶ - 7*10⁵⁸¹⁰⁰ - 1.^[3]