Carmichael's theorem, named for the American mathematician R. D. Carmichael, is a result in number theory about Lucas sequences. It states that for any nondegenerate Lucas sequence of the first kind with relatively prime parameters and positive discriminant, every term beyond the sixth has at least one prime divisor that does not divide any earlier term, with a single exception at the twelfth term. In particular, for n greater than 12, the nth Fibonacci number has at least one prime divisor that does not divide any smaller Fibonacci number. Carmichael proved the theorem in 1913; a simpler proof was later given by Yabuta in 2001, and Bilu, Hanrot, Voutier and Mignotte extended the result to negative discriminants that same year. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor any nondegenerate Lucas sequence of the first kind with relatively prime parameters and positive discriminant, an element Un with n not 1, 2, 6 has at least one prime divisor that does not divide any earlier one, except the 12th Fibonacci number. 1 Classification
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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.
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Source Carmichael's theorem (Wikipedia)
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1. Carmichael's theorem (Wikipedia)
Introduction
has at least one prime divisor that does not divide any earlier one
Introduction [proof-year]
Carmichael (1913, Theorem 21) proved this theorem
In Branch: Number Theory, Lead sentence
In number theory, Carmichael's theorem, named after the American mathematician R.
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