Conjectures
Legendre's Conjecture
luh-ZHAHN-druh
Number Theory
Citation Formats
General Reference
APA Style
BibTeX
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.
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 Cross-Tradition Connections
In Branch
Posed By
Legendre proposed the conjecture in 1808 in his Essai sur la Theorie des Nombres.
Sources
1. Legendre's Conjecture (Wikipedia)
Wikimedia FoundationFirst paragraphQuote, First paragraph
There is a prime number between n² and (n+1)² for every positive integer n.
1. Legendre's Conjecture (Wikipedia)
Wikimedia FoundationPartial Results sectionQuote, 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.
1. Legendre's Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Classification paragraphQuote, In Branch: Number Theory, Classification paragraph
The conjecture is one of Landau's problems (1912) on prime numbers
1. Legendre's Conjecture (Wikipedia)
Wikimedia FoundationPosed By: Adrien-Marie Legendre, Opening paragraphQuote, Posed By: Adrien-Marie Legendre, Opening paragraph
proposed by Adrien-Marie Legendre
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)
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.
View At A Past Year
The atlas records no dated fact of its own for this entry, so there is no other year to choose.