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Wolstenholme's Theorem

If p is a prime >3, then the numerator of the harmonic number

H_(p-1)=1+1/2+1/3+...+1/(p-1)
(1)

is divisible by p^2 and the numerator of the generalized harmonic number

H_(p-1,2)=1+1/(2^2)+1/(3^2)+...+1/((p-1)^2)
(2)

is divisible by p. The numerators of H_(p-1,2) are sometimes known as Wolstenholme numbers.

These imply that if p>=5 is prime, then

(2p-1; p-1)=1 (mod p^3).
(3)

See also

Harmonic Number, Wolstenholme Number, Wolstenholme Prime

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References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 85, 1994.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 21, 1989.

Referenced on Wolfram|Alpha

Wolstenholme's Theorem

Cite this as:

Weisstein, Eric W. "Wolstenholme's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WolstenholmesTheorem.html

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