Wolstenholme prime

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In number theory, a Wolstenholme prime is a certain kind of prime number. A prime p is called a Wolstenholme prime iff the following condition holds:

$\{\{2p-1\}choose\{p-1\}\}\; equiv\; 1\; pmod\{p^4\}.$

Wolstenholme primes are named after Joseph Wolstenholme who proved Wolstenholme's theorem, the equivalent statement for p³ in 1862, following Charles Babbage who showed the equivalent for p² in 1819.

The only known Wolstenholme primes so far are 16843 and 2124679 ; any other Wolstenholme prime must be greater than 10⁹. This data is consistent with the heuristic that the residue modulo p⁴ is a pseudo-random multiple of p³. This heuristic predicts that the number of Wolstenholme primes between K and N is roughly ln ln N - ln ln K. The Wolstenholme condition has been checked up to 10⁹, and the heuristic says that there should be roughly one Wolstenholme prime between 10⁹ and 10²⁴.

See also

• Wieferich prime
• Wilson prime
• Wall-Sun-Sun prime
• List of special classes of prime numbers

References

J. Wolstenholme, "On certain properties of prime numbers", Quarterly Journal of Mathematics 5 (1862), pp. 35–39.

External links

• The Prime Glossary: Wolstenholme prime
• Status of the search for Wolstenholme primes

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Last updated on Saturday March 01, 2008 at 22:56:09 PST (GMT -0800)
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