Theorem

Zsigmondy's Theorem

Number Theory

Zsigmondy's Theorem states that for coprime positive integers a greater than b, and for almost every positive integer n, the number a to the n minus b to the n has a prime factor that divides none of the earlier terms a to the k minus b to the k for k less than n, with only a short, fully classified list of exceptions. Named for Karl Zsigmondy, it is a classical result of number theory guaranteeing so-called primitive prime divisors for these sequences except in the few known exceptional cases.

Facts

Statement

If a > b > 0 are coprime integers, then for any integer n >= 1 there is a prime dividing a^n - b^n but no a^k - b^k for k < n, with finitely listed exceptions. 1

Proof Year

1892 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Connections

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Existence Theorem

Inequality

In Branch

Number Theory

Source Zsigmondy's theorem (Wikipedia)

Sources

1. Zsigmondy's theorem (Wikipedia)

History
The theorem was discovered by Karl Zsigmondy who proved it in 1892.
In Branch: Number Theory, Lead sentence
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any int

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Zsigmondy's Theorem

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