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

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Inequality

In Branch

Number Theory

Source Zsigmondy's theorem (Wikipedia)

Sources

1. Zsigmondy's theorem (Wikipedia)

Statement
there is a prime number p (called a primitive prime divisor) that divides a^n − b^n and does not divide a^k − b^k for any positive integer k < n
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

Has Statement Form

In Branch