If p is an odd prime for which 2p plus 1 is also prime (a Sophie Germain prime condition), then the first case of Fermat's Last Theorem holds for exponent p: there is no solution in integers coprime to p. It was Sophie Germain's major partial contribution toward Fermat's Last Theorem, developed in correspondence with Gauss and Legendre.
Facts
StatementIf an odd prime p has an auxiliary prime q such that no two nonzero pth powers differ by 1 modulo q and p is itself not a pth power modulo q, then at least one of x, y, z in x^p + y^p = z^p must be divisible by p^2, so the first case of Fermat's Last Theorem holds for p. 1 Classification
Statement Form Statement Form Statement Form Connections
Has Statement Form
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.
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.
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.
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.
In Branch
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Proved By
Sources
1. Sophie Germain's theorem (Wikipedia)
Wikimedia FoundationHistory section, first sentence
Germain identified such an auxiliary prime q for every prime less than 100.
History section, second sentence
The theorem and its application to primes p less than 100 were attributed to Germain by Adrien-Marie Legendre in 1823.
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.