The Brun-Titchmarsh Theorem gives an explicit upper bound on the number of primes that can appear in an arithmetic progression up to a given size, bounding that count in terms of the length of the progression and its modulus. Named for Viggo Brun and Edward Charles Titchmarsh, it strengthens the sieve methods Brun pioneered and is a standard tool of analytic number theory for controlling how densely primes can be distributed within a progression.
Facts
Partially Attested
Proof YearThe History section credits Montgomery and Vaughan with proving the sharp bound by sieve methods and the References list their 1973 paper The large sieve; the History text itself gives no year, and the earlier weaker Brun-Titchmarsh version is undated. StatementLet pi(x;q,a) count the primes p congruent to a modulo q with p <= x. Then pi(x;q,a) <= 2x / (phi(q) log(x/q)) for all q < x, an upper bound on the distribution of primes in arithmetic progression. 2 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 Brun-Titchmarsh theorem (Wikipedia)
Sources
1. Wikipedia: Brun-Titchmarsh theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In analytic number theory, the Brun-Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression.
2. Brun-Titchmarsh theorem (Wikipedia)
Statement
\pi (x;q,a)\leq {2x \over \varphi (q)\log(x/q)}
References
Montgomery, H.L.; Vaughan, R.C. (1973)
- In Branch: Analytic Number Theory, Lead sentence
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