The reciprocals of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s 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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

proved in 1737. As

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

, the reciprocals of primes have

repeating decimal A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that i ...

representations. In his later years,

George Salmon George Salmon (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 life to theol ...

(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 James Whitbread Lee Glaisher (5 November 1848, in Lewisham — 7 December 1928, in Cambridge) was a prominent English mathematician and astronomer. He is known for Glaisher's theorem, an important result in the field of integer partitions, a ...

in 1878. For a prime , the period of its reciprocal divides . The sequence of recurrence periods of the reciprocal primes appears in the 1973 Handbook of Integer Sequences.

List of reciprocals of primes

''*''

Full reptend prime In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat ...

s are italicised.
† Unique primes are highlighted.

Full reptend primes

A ''full reptend prime'', ''full repetend prime'', ''proper prime''Dickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or ''long prime'' in base ''b'' is an odd

''p'' such that the

Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as :q_p(a) = \frac, or :\delta_p(a) = \frac. This article is about the former; for the latter see ''p''-derivation. The quotient is named a ...

q_p(b) = \frac

(where ''p'' does not divide ''b'') gives a

cyclic number A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 : ...

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 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. 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 Samuel Yates (May 10, 1919 in Savannah, Georgia – April 22, 1991 in New Brunswick, New Jersey) was a computer engineer and mathematician who first described unique primes in 1980. In 1984 he began the list of "Largest Known Primes" (today The ...

in 1980. A prime number ''p'' is unique if and only if there exists an ''n'' such that :

\frac

is a power of ''p'', where

\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 prim ...

evaluated at

b

. The value of ''n'' is then the period of the decimal expansion of 1/''p''. At present, more than fifty decimal 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 co ...

s 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, ... .

References

External links

* {{Prime number classes Prime numbers Rational numbers