Proposed by Adrien-Marie Legendre in his 1808 work on number theory, this conjecture asks whether a prime can always be found between consecutive perfect squares. It is one of the four basic problems about the distribution of primes that Edmund Landau named as unapproachable in his 1912 address, and none of the four has been resolved since. 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
StatementThere is a prime number between n squared and (n plus one) squared, for every positive integer n. 1 Proposed Year Prize StatusNot a Millennium Prize Problem; no institutional cash prize is attached. It is one of Landau's problems, a standing list of number theory questions considered basic and unapproachable since 1912. 2 Progress Toward ResolutionNo proof or counterexample exists. The claim has been checked by computer for enormous ranges of n, and results have been proved for closely related but weaker statements, such as a prime always occurring between x and x plus x to the twenty one fortieths power for large x, but the original conjecture about consecutive squares remains completely open. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Legendre's Conjecture (Wikipedia)
Open Questions
Source Legendre's Conjecture (Wikipedia)
Posed By
Legendre proposed the conjecture in 1808 in his Essai sur la Theorie des Nombres.
Source Legendre's Conjecture (Wikipedia)
Sources
1. Legendre's Conjecture (Wikipedia)
Wikimedia FoundationFirst paragraph
There is a prime number between n² and (n+1)² for every positive integer n.
Partial Results section
Baker, Harman, and Pintz proved that there is a prime in the interval [x−x^(21/40), x] for all large x.
- Lead section
In Branch: Number Theory, Classification paragraph
The conjecture is one of Landau's problems (1912) on prime numbers
Posed By: Adrien-Marie Legendre, Opening paragraph
proposed by Adrien-Marie Legendre
View the Source 2. Prime Numbers (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsSome unsolved problems sectionQuote, Some unsolved problems section
Is there always a prime between n² and (n + 1)²?
Open Questions (1 open question)
Is there really always at least one prime number between n squared and (n plus one) squared, for every positive integer n?
No proof or counterexample has ever been found, despite the claim being checked by computer for enormous ranges of n. Results exist for closely related but weaker statements, such as a prime between consecutive cubes for large enough n, but nobody has found a way to close the gap for consecutive squares, which is exactly the gap that makes it one of Landau's four unapproachable problems named in 1912.
What would resolve this A general proof covering every positive integer n, or a single confirmed counterexample: one gap between consecutive squares, however large, that a careful search shows contains no prime at all.
Analytic number theoryLegendre's Conjecture (Wikipedia)
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