mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Wolstenholme number is a

number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

that is the

numerator A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...

of the

generalized harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...

''H''_''n'',2. The first such numbers are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... . These numbers are named after

Joseph Wolstenholme Joseph Wolstenholme (30 September 1829 – 18 November 1891) was an United Kingdom, English mathematician. Wolstenholme was born in Eccles, Greater Manchester, Eccles near Salford, Greater Manchester, Salford, Lancashire, England, the son of a M ...

, who proved

Wolstenholme's theorem In mathematics, Wolstenholme's theorem states that for a prime number ''p'' ≥ 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a mult ...

on modular relations of the generalized harmonic numbers.

References

* Integer sequences {{Num-stub