Wolstenholme theorem

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2020 Mathematics Subject Classification: Primary: 11A07 [MSN][ZBL]

Let $p$ be a prime number greater than 3. The numerator of the fraction $$ \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{p-1} $$ is divisible by $p^2$.

An equivalent form of the theorem is that if $x^*$ denotes the solution to the equation $x x^* \equiv 1 \pmod {p^2}$ then $$ 1^* + 2^* + \cdots + (p-1)^* \equiv 0 \pmod{p^2} \ . $$


  • G. H. Hardy, E. M. Wright,(with R. Heath-Brown, J. Silverman) "An Introduction to the Theory of Numbers" (6th ed.) Oxford University Press (2008) ISBN 0-19-921986-9 Zbl 1159.11001
  • N. Rama Rao, "Some congruences modulo $m$" Bull. Calcutta math. Soc. 29 (1938) 167-170 Zbl 64.0097.02
  • J. Wolstenholme, "On certain properties of prime numbers", Quart. J. Math. 5 (1862), 35-99
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