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. 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 first few emirps are
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, ... (sequence A006567 in the OEIS). The first few reversible primes are
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, ... (sequence A007500 in the OEIS). 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 10 n − 1 {\displaystyle 10^{n}-1} being a multiple of 9 for every non-negative integer n {\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. The integer's reverse is 10111956 - 7 × 1058100 - 1.
History The term "emirp" was coined by American mathematician Jeremiah Farrell.
Count of emirps The following table shows the number of n-digit emirps and reversible primes: (sequences A152014 and A048054 in the OEIS)
The number of n-digit reversible primes is equal to the number of n-digit emirps plus the number of n-digit palindromic primes.
Largest known emirps The following table shows the largest known emirps throughout the years:
Other bases Emirps also exist for bases other than 10. For example, 101112 = 23 is an emirp in the binary numeral system because 23 is a prime and its reverse 111012 = 29 is also a prime. The first few binary emirps are
11, 13, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 83, 97, 101, ... (sequence A080790 in the OEIS). Dartyge et al. (2023) have shown that, in big-O notation, the number of n-digit reversible primes in binary is O ( 2 n n 2 ) {\displaystyle O\left({\frac {2^{n}}{n^{2}}}\right)} . They also conjecture that the number of n-digit reversible primes in binary is ( 3 + o ( 1 ) ) 2 n − 1 ( log 2 n ) 2 {\displaystyle (3+o(1)){\frac {2^{n-1}}{(\log 2^{n})^{2}}}} as n goes large.
Emirpimes The term emirpimes (singular) is used also in places to treat semiprimes in a similar way. That is, an emirpimes is a semiprime that is also a (distinct) semiprime upon reversing its digits. The first few emirpimeses are
15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (sequence A097393 in the OEIS).
References