Wolstenholme's theorem

The set of Wolstenholme (after Joseph Wolstenholme ) is a statement from the mathematical branch of number theory. In one possible form it says:

Is a prime number, as is the numerator of the rational number

Divisible by.

Examples, other formulations conclusions

The set can be well illustrated by a few examples. Consider:

  • P = 7, then, and the counter is obviously divisible by.
  • P = 13, then. The counter 86021 must now be after the sentence divisible by:

The set of Wolstenholme is equivalent to saying that the counter of

Is divisible by.

One conclusion from the set is the congruence

Also in the form of

Can be written.

Wolstenholme primes

A Wolstenholme prime p is a prime number that satisfies a stronger version of the theorem of Wolstenholme, more precisely meets the one of the following equivalent conditions:

  • The numerator of
  • The numerator of
  • It is the congruence
  • The numerator of the Bernoulli number is divisible by.

The two previously only known Wolstenholme primes are 16843 ( Selfridge and Pollack 1964) and 2,124,679 ( Buhler, Crandall, Ernvall and Metsänkylä 1993). Each additional Wolstenholme prime number must be greater than 109. It has been conjectured that an infinite number of Wolstenholme primes, approximately below ( McIntosh 1995).

Related term

Looking only at odd denominator, ie the sum

For a prime number, then the counter is exactly divisible by then, if the stronger form

The set of Euler -Fermat theorem applies. Such primes are called Wieferich primes.

History

From the set of Wilson the congruence follows

For every prime number p and any natural number n

Charles Babbage in 1819 proved the congruence

For every prime p> 2

Joseph Wolstenholme proved 1862, the congruence

For every prime p> 3

Modular arithmetic John Selfridge Richard Crandall Wilson's theorem G. H. Hardy E. M. Wright Wolfram Research
710739
de