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

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

proved in 1737. Like all

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

, the reciprocals of primes have

repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...

representations. In his later years,

George Salmon George Salmon FBA FRS FRSE (25 September 1819 – 22 January 1904) was a distinguished and influential Irish mathematician and Anglican theologian. After working in algebraic geometry for two decades, Salmon devoted the last forty years of his ...

(1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes. Contemporaneously,

William Shanks William Shanks (25 January 1812 – June 1882) was an English amateur mathematician. He is famous for his calculation of '' '' (pi) to 707 places in 1873, which was correct up to the first 527 places. The error was discovered in 1944 by D. F. Fe ...

(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 James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher and Cecilia Glaisher, was a prolific English mathematician and astronomer. His large collection of (mostly) English ...

in 1878. For a prime , the period of its reciprocal will be equal to or will divide . The sequence of recurrence periods of the reciprocal primes appears in the 1973 Handbook of Integer Sequences.

Unique primes

A prime ''p'' ≠ 2, 5 is called unique if there is no other prime ''q'' such that the

period length A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

of the decimal expansion of its

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

, 1 / ''p'', is equal to the period length of the reciprocal of ''q'', 1 / ''q''. 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. At present, more than fifty unique primes or

probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...

s are known. However, there are only twenty-three unique primes below 10¹⁰⁰. contains a list of unique primes and are those primes ordered by period length; contains periods (ordered by corresponding primes) and contains periods, sorted, corresponding with A007615. the repunit (10^8177207 – 1)/9 is the largest known probable unique prime. In 1996 the largest ''proven'' unique prime was (10¹¹³² + 1)/10001 or, using the notation above, (99990000)₁₄₁ + 1. It has 1128 digits. The record has been improved many times since then. the largest proven unique prime is

\Phi_(-100)

, it has 23732 digits. Here

\Phi_n(b)

denotes the

n

cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primiti ...

evaluated at

b

.''The Top Twenty Unique''; Chris Caldwell
/ref>

References

{{Prime number classes Prime numbers Rational numbers