Zsigmondy's theorem

For coprime integers a > b > 0 and n > 1, aⁿ − bⁿ has a prime divisor that divides no earlier difference aᵏ − bᵏ, except when n = 2 and a + b is a power of 2, or when (a, b, n) = (2, 1, 6).

Cyclotomic polynomial: A polynomial whose roots are the primitive roots of unity of a given order. Factoring aⁿ − bⁿ into cyclotomic values isolates the factors associated with exponent n.
Coprime integers: Integers whose greatest common divisor is 1. The theorem's hypothesis prevents shared prime factors from trivially persisting through every difference.
Karl Zsigmondy: An Austrian mathematician who proved a theorem on new prime divisors of differences of powers in 1892. His result gives the theorem its name and establishes its general exceptional cases.
Repunit: An integer whose digits are all 1 in a fixed numeral base. Repunit factorizations reduce to differences of powers, where new prime divisors can be guaranteed.
Schur's theorem: A theorem stating that every nonconstant integer polynomial has infinitely many prime divisors among its values. Schur guarantees infinitely many primes across polynomial values, rather than a new divisor at each exponent.
Multiplicative order: The least positive exponent that raises a group element to the identity. A prime's first appearance is governed by the order of a times b⁻¹ modulo that prime.
Divisibility: The relation in which one integer is an exact integer multiple of another. The theorem compares which primes divide the nth difference and its predecessors.
Bang's theorem: A result guaranteeing primitive prime divisors for many terms of sequences formed from powers. Bang proved an earlier, narrower case for differences involving 1 and a larger integer.
Cyclotomic polynomial values: Integers obtained by evaluating cyclotomic polynomials at specified integer arguments. The theorem supplies prime divisors of these values with prescribed order properties.
Bilu–Hanrot–Voutier theorem: A theorem guaranteeing primitive prime divisors for Lucas and Lehmer sequences beyond a fixed index. It extends primitive-divisor methods to broader recurrence sequences instead of just aⁿ − bⁿ.

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Goormaghtigh conjectureRelated: Primitive prime divisors can limit how two repunit factorizations overlap.