Wolstenholme's Theorem

The set of Wolstenholme (by Joseph Wolstenholme ) is a statement from the mathematical branch of number theory . It reads:

If a prime number , then has the harmonic number ${\displaystyle p\geq 5}$

${\displaystyle H(p-1)=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\ldots +{\frac {1}{p-1}}}$

a numerator that can be divided by (in completely abbreviated form and therefore also in any other representation as the quotient of two whole numbers ). ${\displaystyle p^{2}}$

Examples, other formulations, conclusions

A few examples to illustrate this:

• ${\displaystyle p=7{\text{:}}\quad 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}+{\tfrac {1}{6}}={\tfrac {49}{20}},}$the counter is divisible by. ${\displaystyle 49=1\cdot 7^{2}}$ ${\displaystyle 7^{2}}$
• ${\displaystyle p=13{\text{:}}\quad 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}+{\tfrac {1}{6}}+{\tfrac {1}{7}}+{\tfrac {1}{8}}+{\tfrac {1}{9}}+{\tfrac {1}{10}}+{\tfrac {1}{11}}+{\tfrac {1}{12}}={\tfrac {86021}{27720}},}$the counter is divisible by. ${\displaystyle 86021=509\cdot 13^{2}}$ ${\displaystyle 13^{2}}$

Wolstenholme's theorem is equivalent to the statement that the numerator of

${\displaystyle 1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\ldots +{\frac {1}{(p-1)^{2}}}}$

is divisible by . ${\displaystyle p}$

One consequence of the theorem is congruence

${\displaystyle {\binom {2p}{p}}\equiv 2\mod {p^{3}},}$

which also in the form

${\displaystyle {\binom {2p-1}{p-1}}\equiv 1\mod {p^{3}}}$

can be written.

Wolstenholme prime numbers

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

• The counter of

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\dots +{\frac {1}{p-1}}}$

is divisible by. ${\displaystyle p^{3}}$

• The counter of

${\displaystyle 1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{(p-1)^{2}}}}$

is divisible by. ${\displaystyle p^{2}}$

• The congruence applies

${\displaystyle {\binom {2p}{p}}\equiv 2\mod {p^{4}}.}$

• The congruence applies

${\displaystyle {\binom {2p-1}{p-1}}\equiv 1\mod {p^{4}}.}$

• The numerator of the Bernoulli number is divisible by. ${\displaystyle B_{p-3}}$ ${\displaystyle p}$

The two only known Wolstenholme prime numbers so far are 16843 ( Selfridge and Pollack 1964) and 2124679 (Buhler, Crandall , Ernvall and Metsänkylä 1993). Every further Wolstenholme prime number would have to be greater than 10 ⁹ . The assumption was made that there are an infinite number of Wolstenholme prime numbers, approximately below (McIntosh 1995). ${\displaystyle \log(\log(x))}$ ${\displaystyle x}$

Related term

If one only considers summands with an odd denominator, i.e. the sum

${\displaystyle 1+{\frac {1}{3}}+{\frac {1}{5}}+\dots +{\frac {1}{p-2}}}$

for a prime number , the numerator is divisible by if and only if the stronger form ${\displaystyle p\geq 3}$ ${\displaystyle p}$

${\displaystyle 2^{p-1}\equiv 1\mod {p^{2}}}$

of the Euler-Fermat theorem holds. Such prime numbers are called Wieferich prime numbers .

history

From the set of Wilson congruence follows

${\displaystyle {\binom {np-1}{p-1}}\equiv 1{\pmod {p}}}$

for every prime and every natural number ${\displaystyle p}$ ${\displaystyle n.}$

Charles Babbage proved the congruence in 1819

${\displaystyle {\binom {2p-1}{p-1}}\equiv 1{\pmod {p^{2}}}}$

for every prime number ${\displaystyle p>2.}$

Joseph Wolstenholme proved the congruence in 1862

${\displaystyle {\binom {2p-1}{p-1}}\equiv 1{\pmod {p^{3}}}}$

for every prime number ${\displaystyle p>3.}$

literature

• GH Hardy , EM Wright : An introduction to the theory of numbers. 6th edition. Oxford University Press, Oxford 2008, ISBN 978-0-19-921985-8 (English; revised by D. R. Heath-Brown and J. H. Silverman ).

Web links

• The Prime Glossary: Wolstenholme prime (English)
• Eric W. Weisstein : Wolstenholme's Theorem . In: MathWorld (English).

Individual evidence

1. ↑ ^{a b} J. Wolstenholme : On certain properties of prime numbers. In: The quarterly journal of pure and applied mathematics 5. 1862, pp. 35-39 (English).
2. ↑ Hardy, Wright: An introduction to the theory of numbers. 2008, p. 112 (English; Theorem 115).
3. ↑ Hardy, Wright: An introduction to the theory of numbers. 2008, p. 114 (English; Theorem 117).
4. ^ Anthony Gardiner: Four problems on prime power divisibility. In: The American Mathematical Monthly December 95 , 1988, pp. 926-931.
5. ^ JL Selfridge , BW Pollack: Fermat's last theorem is true for any exponent up to 25,000. In: Notices of the AMS 11.1964, p. 97 (English; abstract only; 16843 not expressly stated).
6. ↑ J. Buhler, R. Crandall , R. Ernvall, T. Metsänkylä: Irregular primes and cyclotomic invariants to four million . In: Mathematics of Computation July 61 , 1993, pp. 151-153 (English).
7. ^ Richard J. McIntosh, Eric L. Roettger: A search for Fibonacci-Wieferich and Wolstenholme primes . (PDF; 151 kB). In: Mathematics of Computation , 76, October 2007, pp. 2087-2094 (English).
8. ^ Richard J. McIntosh: On the converse of Wolstenholme's theorem . (PDF; 190 kB). In: Acta Arithmetica , 71, 1995, pp. 381-389 (English).
9. ↑ Hardy, Wright: An introduction to the theory of numbers. 2008, p. 135 (English; Theorem 132).
10. ^ Charles Babbage : Demonstration of a theorem relating to prime numbers. In: The Edinburgh philosophical journal 1. 1819, pp. 46–49 (English; “n + 1.n + 2.n + 3 ...” means “(n + 1) (n + 2) (n + 3 ) ... "; the reverse is also asserted:" otherwise it is not ", but not proven and is wrong for squares of Wolstenholme prime numbers).