Theorem

Linnik's Theorem

Number Theory

Linnik's Theorem answers a natural question left open by Dirichlet's Theorem on Arithmetic Progressions, showing that the least prime number appearing in an arithmetic progression of common difference d is bounded above by a constant multiple of d raised to a fixed power, rather than growing without any controlled rate as d increases. Named for Yuri Vladimirovich Linnik, who proved it in 1944, the theorem established that both the constant and the exponent are computable in principle, though Linnik's own proof did not supply their numerical values.

Facts

Statement

There exist positive constants c and L such that the least prime p(a,d) in the arithmetic progression a + nd, for coprime positive integers a and d with 1 <= a <= d - 1, satisfies p(a,d) < c d^L. 1

Proof Year

1944 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

Analytic Number Theory

Source Linnik's theorem (Wikipedia)

Sources

1. Linnik's theorem (Wikipedia)

Introduction, sentence 1
It asserts that there exist positive c and L such that, if we denote p(a,d) the least prime in the arithmetic progression
Introduction, sentence 2
who proved it in 1944
In Branch: Analytic Number Theory, Lead sentence
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions.

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Linnik's Theorem

Has Statement Form

In Branch