Vipul: Created page with "==Statement== ===Statement in terms of near-central binomial coefficients=== Suppose $p$ is a prime number greater than 3. Then, we have the following congrue..."

2012-01-03T20:35:50Z

Created page with "==Statement== ===Statement in terms of near-central binomial coefficients=== Suppose <math>p</math> is a prime number greater than 3. Then, we have the following congrue..."

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==Statement==

===Statement in terms of near-central binomial coefficients===

Suppose <math>p</math> is a [[prime number]] greater than 3. Then, we have the following congruence condition on the [[fact about::binomial coefficient]]:

<math>\binom{2p - 1}{p - 1} \equiv 1 \pmod {p^3}</math>

===Statement in terms of prime cancellation from binomial coefficient===

Suppose <math>p</math> is a [[prime number]] greater than 3. Then, we have the following congruence condition:

<math>\binom{ap}{bp} \equiv \binom{a}{b} \pmod{p^3}</math>

This version is due to Glaisher and is similar in structure to [[Lucas' theorem]].

===Statement in terms of generalized harmonic numbers===

The following two congruences hold:

<math>\! 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{p-1} \equiv 0 \pmod{p^2}</math>

and

<math>\! 1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{(p-1)^2} \equiv 0 \pmod{p}</math>

What we mean by this is that if we simplify the expressions on the left side and bring them into fractions in reduced form, the numerators are divisible by <math>p^2</math> and <math>p</math> respectively.

Wolstenholme's theorem - Revision history

Vipul: Created page with "==Statement== ===Statement in terms of near-central binomial coefficients=== Suppose p is a prime number greater than 3. Then, we have the following congrue..."

Vipul: Created page with "==Statement== ===Statement in terms of near-central binomial coefficients=== Suppose $p$ is a prime number greater than 3. Then, we have the following congrue..."