The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737.

As rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes.

Contemporaneously, William Shanks (1812–1882) calculated numerous reciprocals of primes and their repeating periods, and published two papers "On Periods in the Reciprocals of Primes" in 1873 and 1874. In 1874 he also published a table of primes, and the periods of their reciprocals, up to 20,000 (with help from and "communicated by the Rev. George Salmon"), and pointed out the errors in previous tables by three other authors.

Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. For a prime p, the period of its reciprocal divides p − 1.

The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.

* Full reptend primes are italicised.
† Unique primes are highlighted.

A full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient

(where p does not divide b) gives a cyclic number with p − 1 digits. Therefore, the base b expansion of ${\displaystyle 1/p}$ repeats the digits of the corresponding cyclic number infinitely.