An emirp (''prime'' spelled backwards) is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that results in a different prime when its decimal digits 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 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, ... . All non-palindromic

permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study the ...

s are emirps. , the largest known emirp is 10¹⁰⁰⁰⁶+941992101×10⁴⁹⁹⁹+1, found by Jens Kruse Andersen in October 2007. The term 'emirpimes' (singular) is used also in places to treat

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...

s in a similar way. That is, an emirpimes is a semiprime that is also a (distinct) semiprime upon reversing its digits. It is an open problem whether there are infinitely many emirps.

Other bases

The emirps in

base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

are (using rotated two and three for ten and eleven, respectively):

15, 51, 57, 5Ɛ, 75, Ɛ5, 107, 117, 11Ɛ, 12Ɛ, 13Ɛ, 145, 157, 16Ɛ, 17Ɛ, 195, 19Ɛ, 1ᘔ7, 1Ɛ5, 507, 51Ɛ, 541, 577, 587, 591, 59Ɛ, 5Ɛ1, 5ƐƐ, 701, 705, 711, 751, 76Ɛ, 775, 785, 7ᘔ1, 7ƐƐ, Ɛ11, Ɛ15, Ɛ21, Ɛ31, Ɛ61, Ɛ67, Ɛ71, Ɛ91, Ɛ95, ƐƐ5, ƐƐ7, ...

Emirps with added mirror properties

There is a subset of emirps ''x'', with mirror ''x_m'', such that ''x'' is the ''y''th prime, and ''x_m'' is the ''y_m''th prime. (E.g. 73 is the 21st prime number; its mirror, 37, is the 12th prime number; 12 is the mirror of 21.)

Twin emirp

A twin emirp (or emirp twin) is a pair of emirp such that the smaller one and its reversal is a

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...

. For example 71 is the smallest twin emirp. 71, 73, 17 and 19 are all different

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, so 71 is a twin emirp. The sequence of twin emirps is 71, 1031, 1151, 1229, 3299, 3371, 3389, 3467, 3851, 7457, 7949, 9011, 9437, 10007, ... (the sequenc
A175215
in the

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

). The largest found Twin emirp is 10⁴⁹⁹ + 174295123052 +/-1. The smallest twin emirp that is sum of first twin emirps is 71 + 1031 + 1151 + ... + 901814489 = 18036881674937 OEIS sequence for twin emirp
equal partial sum of twin emirps
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References

Classes of prime numbers Base-dependent integer sequences {{Num-stub