Thereciprocals ofprime numbers have been of interest to mathematicians for various reasons. Theydo not have a finite sum, asLeonhard Euler proved in 1737.

Asrational numbers, the reciprocals of primes haverepeating decimal representations. In his later years,George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes.^[1]

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^[2] and 1874.^[3] 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.^[4]

The last part of Shanks's 1874 table of primes and their repeating periods. In the top row, 6952 should be 6592 (the error is easy to find, since the period for a primep must dividep − 1). In his report extending the table to 30,000 in the same year, Shanks did not report this error, but he reported that in the same column, opposite 19841, the 1984 should be 64. *Another error which may have been corrected since his work was published is opposite 19423—the reciprocal repeats every 6474 digits, not every 3237.

Rules for calculating the periods of repeating decimals from rational fractions were given byJames Whitbread Lee Glaisher in 1878.^[5] For a primep, the period of its reciprocal dividesp − 1.^[6]

The sequence of recurrence periods of the reciprocal primes (sequenceA002371 in theOEIS) appears in the 1973 Handbook of Integer Sequences.

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

Afull reptend prime,full repetend prime,proper prime^[7]: 166 orlong prime inbaseb is anoddprime numberp such that theFermat quotient

${\displaystyle q_p(b)=\frac{b^{p-1}-1}{p}}$

(wherep does notdivideb) gives acyclic number withp − 1 digits. Therefore, the baseb expansion of${\displaystyle 1/p}$ repeats the digits of the corresponding cyclic number infinitely.

A primep (wherep ≠ 2, 5 when working in base 10) is calledunique if there is no other primeq such that theperiod length of the decimal expansion of itsreciprocal, 1/p, is equal to the period length of the reciprocal ofq, 1/q.^[8] For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667). Unique primes were described bySamuel Yates in 1980.^[9] A prime numberp is unique if and only if there exists ann such that

${\displaystyle \frac{{\mathrm{\Phi }}_n(10)}{gcd({\mathrm{\Phi }}_n(10),n)}}$

is a power ofp, where${\displaystyle {\mathrm{\Phi }}_n(b)}$ denotes the${\displaystyle n}$thcyclotomic polynomial evaluated at${\displaystyle b}$. The value ofn is then the period of the decimal expansion of 1/p.^[10]

At present, more than fifty decimal unique primes orprobable primes are known. However, there are only twenty-three unique primes below 10¹⁰⁰.

The decimal unique primes are

3, 11, 37, 101, 9091, 9901, 333667, 909091, ... (sequenceA040017 in theOEIS).

• Parker, Matt (March 14, 2022)."The Reciprocals of Primes - Numberphile".YouTube.

• Prime numbers
• Rational numbers

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