Conjectures
Twin Prime Conjecture
TWIN PRYM kon-JEK-cher
Number Theory
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Concerns twin primes, pairs of primes that differ by exactly two, such as (3, 5), (11, 13) and (10,006,427, 10,006,429). Euclid's proof that primes themselves never run out is ancient, but whether the much rarer twin PAIRS also never run out was first clearly posed in its modern form by the French mathematician Alphonse de Polignac in 1849 (not yet a live entity on this atlas), as a special case of a broader conjecture about prime gaps of any fixed even size. A major advance came in 2013, when Yitang Zhang proved that some fixed gap smaller than 70 million recurs infinitely often, a breakthrough rapidly improved by other mathematicians' collaborative Polymath project to a gap of 246; the full conjecture, a gap of exactly two, remains unproven.
Facts
StatementThere are infinitely many prime numbers p such that p plus two is also prime. 1 Proposed YearFirst clearly posed in its modern form by Alphonse de Polignac in 1849, as a special case of a broader conjecture about fixed prime gaps; not yet a live entity on this atlas. Prize StatusNot a Millennium Prize Problem; no major institutional cash prize is offered for its proof. 1 Progress Toward ResolutionYitang Zhang proved in 2013 that some fixed gap below 70 million recurs between consecutive primes infinitely often, the first proof of any bounded gap; the Polymath8 collaborative project rapidly sharpened this to a gap of 246. The full conjecture, a gap of exactly two, remains unproven. 1 Cross-Tradition Connections
Associated With
Euclid, Mathematicians Euclid's proof that primes never run out is the ancient ancestor result; the modern twin-pair form is a much later, separate conjecture.
The atlas's own still-open conjecture about pairs of primes two apart.
In Branch
Posed By
First clearly posed the modern form in 1849, as the k=1 case of his own broader conjecture about fixed prime gaps.
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statistics
Wikipedia: Twin Prime
Wikimedia FoundationTwin prime conjectureQuote, Twin prime conjecture
It is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair.
Wikipedia: Twin Prime
Wikimedia FoundationPosed By: Alphonse de Polignac, Twin prime conjecture sectionQuote, Posed By: Alphonse de Polignac, Twin prime conjecture section
In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case k = 1 of de Polignac's conjecture is the twin prime conjecture.
Wikipedia: Twin Prime
Wikimedia FoundationIn Branch: Number Theory, Opening paragraphQuote, In Branch: Number Theory, Opening paragraph
The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years.
Open Questions (1 open question)
Do infinitely many twin prime pairs really exist, or does the last one eventually appear?
2013's breakthrough (Yitang Zhang, later sharpened by the Polymath collaborative project to a gap of 246) proved infinitely many prime pairs exist within SOME bounded gap, but closing that gap all the way down to exactly two, the original conjecture, remains open.
What would resolve this A proof narrowing the proven bounded gap (currently 246) all the way to 2, or a fundamentally different argument establishing the gap-2 case directly.
Analytic number theoryMacTutor History of Mathematics Archive
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