Reciprocals of primes

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 all 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]

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 prime p must divide p − 1). In his report extending the table to 30,000 in the same year, Shanks did not report this error, but 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 by James Whitbread Lee Glaisher in 1878.[5] For a prime p, the period of its reciprocal will be equal to or will divide p − 1.[6]

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

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.[7] 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. Unique primes were described by Samuel Yates in 1980.[8]

At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100. A040017 contains a list of unique primes and A007615 are those primes ordered by period length; A051627 contains periods (ordered by corresponding primes) and A007498 contains periods, sorted, corresponding with A007615.

As of 2021 the repunit (108177207 – 1)/9 is the largest known probable unique prime.[9]

In 1996 the largest proven unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141 + 1. It has 1128 digits.[10] The record has been improved many times since then. As of 2021 the largest proven unique prime is , it has 23732 digits. Here denotes the th cyclotomic polynomial evaluated at .[11]

References
  1. "Obituary Notices – George Salmon". Proceedings of the London Mathematical Society. Second Series. 1: xxii–xxviii. 1904. Retrieved 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 (1873). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics. II: 41–43. Retrieved 27 March 2022.
  3. Shanks, William (1874). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics. III: 52–55. Retrieved 27 March 2022.
  4. Shanks, William (1874). "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. 22: 200–210. Retrieved 27 March 2022.
  5. Glaisher, J. W. L. (1878). "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. 3 (V): 185–206. Retrieved 27 March 2022.
  6. Cook, John D. "Reciprocals of primes". johndcook.com. Retrieved 6 April 2022.
  7. Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.
  8. Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.
  9. PRP Records: Probable Primes Top 10000
  10. "Wolfram Alpha". Wolfram Alpha. Retrieved 8 June 2023.
  11. The Top Twenty Unique; Chris Caldwell
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