Emirps

An emirp is a type of prime number that, when its digits are reversed, still results in a prime number, but the reversed number must be different from the original one. The word "emirp" comes from the word "prime" spelled backwards.

Emirps are part of a special group of numbers called "reversible primes." They are studied in number theory, a branch of mathematics.

Example

• 13 is an emirp because when reversed, it becomes 31, which is also a prime number.
• 17 is an emirp because its reverse, 71, is also prime.

History

The word "emirp" was invented by the American mathematician Jeremiah Farrell.

Properties

• An emirp always has a minimum of two digits.
• An emirp is not a palindrome (a number that is the same forwards and backwards). For example, 131 is not an emirp because it is the same when reversed.
• The smallest emirp is 13.
• The largest known emirp as of 2026 is 3867632931*10^10001 + 1, which was found by Stephan Schöler

List of emirp pairs

(13, 31), (17, 71), (37, 73), (79, 97), (107, 701), (113, 311), (149, 941), (157, 751), (179, 971), (199, 991), (337, 733), (347, 743), (359, 953), (389, 983), (709, 907), (739, 937), (769, 967), (1009, 9001), (1021, 1201), (1031, 1301), (1033, 3301), (1061, 1601), (1069, 9601), (1091, 1901), (1097, 7901), (1103, 3011), (1109, 9011), (1151, 1511), (1153, 3511), (1181, 1811), (1193, 3911), (1213, 3121), (1217, 7121), (1223, 3221), (1229, 9221), (1231, 1321), (1237, 7321), (1249, 9421), (1259, 9521), (1279, 9721), (1283, 3821), (1381, 1831), (1399, 9931), (1409, 9041), (1429, 9241), (1439, 9341), (1453, 3541), (1471, 1741), (1487, 7841), (1499, 9941), (1523, 3251), (1559, 9551), (1583, 3851), (1597, 7951), (1619, 9161), (1657, 7561), (1669, 9661), (1723, 3271), (1733, 3371), (1753, 3571), (1789, 9871), (1847, 7481), (1867, 7681), (1879, 9781), (1913, 3191), (1933, 3391), (1949, 9491), (1979, 9791), etc...

References

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