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Wolstenholme's theorem [wikipedia] - mklabgr - 06-20-2026
Wolstenholme's theorem [wikipedia]
Summary
Wolstenholme’s theorem is a result in number theory stating that for every prime number $(p \ge 5)$, the binomial coefficient ($\binom{2p-1}{p-1}$) satisfies the congruence $(\binom{2p-1}{p-1}\equiv1\pmod{p^3})$. It was proved by Joseph Wolstenholme in 1862 and has equivalent forms involving harmonic sums, such as $(1+\frac12+\frac13+\cdots+\frac1{p-1}\equiv0\pmod{p^2})$ and $(1+\frac1{2^2}+\frac1{3^2}+\cdots+\frac1{(p-1)^2}\equiv0\pmod p)$.
The theorem connects combinatorics, prime numbers, and modular arithmetic, and it has inspired many generalizations. A stronger case, where the congruence holds modulo (p^4), defines special primes called Wolstenholme primes; only a few are known, and it remains an active area of research in number theory.
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