Reciprocals of primes explained

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.

Like 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.^[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]

Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878.^[5] For a prime, the period of its reciprocal divides .^[6]

The sequence of recurrence periods of the reciprocal primes appears in the 1973 Handbook of Integer Sequences.

List of reciprocals of primes

Prime (p)	Period length	Reciprocal (1/p)
2	0	0.5
3	† 1	0.
5	0	0.2
7	6	0.
11	† 2	0.
13	6	0.
17	16	0.
19	18	0.
23	22	0.
29	28	0.
31	15	0.
37	† 3	0.
41	5	0.
43	21	0.
47	46	0.
53	13	0.
59	58	0.
61	60	0.
67	33	0.
71	35	0.
73	8	0.
79	13	0.
83	41	0.
89	44	0.
97	96	0.
101	† 4	0.
103	34	0.
107	53	0.
109	108	0.
113	112	0.
127	42	0.

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

Full reptend primes

See main article: Full reptend prime. A full reptend prime, full repetend prime, proper prime^[7] or long prime in base b is an odd prime number p such that the Fermat quotient

q _{p(b)} = (b ^{p} -1)/(p)

(where p does not divide b) gives a cyclic number with p - 1 digits. Therefore, the base b expansion of

1 /p repeats the digits of the corresponding cyclic number infinitely.

Unique primes

A prime p (where p ≠ 2, 5 when working in base 10) is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 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 by Samuel Yates in 1980.^[9] A prime number p is unique if and only if there exists an n such that

(\Phi _{n(10)})/(\gcd(\Phi _{n(10),} n)) is a power of p, where \Phi _{n(b)} denotes the n th cyclotomic polynomial evaluated at b . The value of n is then the period of the decimal expansion of 1/p.^[10]

At present, more than fifty decimal unique primes or probable 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, ... .

External links

• Web site: Parker . Matt . March 14, 2022 . The Reciprocals of Primes - Numberphile . YouTube.

Notes and References

1. Obituary Notices – George Salmon . Proceedings of the London Mathematical Society . Second Series . 1904 . 1 . xxii–xxviii . 27 March 2022 . ...there was one branch of calculation which had a great fascination for him. It was the determination of the number of figures in the recurring periods in the reciprocals of prime numbers..
2. Shanks . William . On Periods in the Reciprocals of Primes . The Messenger of Mathematics . 1873 . II . 41–43 . 27 March 2022.
3. Shanks . William . On Periods in the Reciprocals of Primes . The Messenger of Mathematics . 1874 . III . 52–55 . 27 March 2022.
4. Shanks . William . On the Number of Figures in the Period of the Reciprocal of Every Prime Number Below 20,000 . Proceedings of the Royal Society of London . 1874 . 22 . 200–210 . 27 March 2022.
5. Glaisher . J. W. L. . On circulating decimals with special reference to Henry Goodwin's 'Table of circles' and 'Tabular series of decimal quotients' . Proceedings of the Cambridge Philosophical Society: Mathematical and physical sciences . 1878 . 3 . V . 185–206 . 27 March 2022.
6. Web site: Cook . John D. . Reciprocals of primes . johndcook.com . 6 April 2022.
7. Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
8. Web site: Caldwell. Chris. Unique prime. The Prime Pages. 11 April 2014.
9. Yates . Samuel . Periods of unique primes . 0445.10009 . Math. Mag. . 53 . 314 . 1980 .
10. Web site: Generalized Unique . Prime Pages . 9 December 2023.