Legendre's conjecture, proposed by
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...
, states that there is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
between
and
for every
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
.
The
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
is one of
Landau's problems (1912) on prime numbers, and is one of many
open problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
s on the spacing of prime numbers.
Prime gaps
If Legendre's conjecture is true, the
gap between any prime ''p'' and the next largest prime would be
, as expressed in
big O notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
. It is one of a family of results and conjectures related to
prime gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e.,
:g_n = p_ - p_n. ...
s, that is, to the spacing between prime numbers. Others include
Bertrand's postulate
In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always at least one ...
, on the existence of a prime between
and
,
Oppermann's conjecture on the existence of primes between
,
, and
,
Andrica's conjecture and
Brocard's conjecture on the existence of primes between squares of consecutive primes, and
Cramér's conjecture that the gaps are always much smaller, of the order
. If Cramér's conjecture is true, Legendre's conjecture would follow for all sufficiently large ''n''.
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statis ...
also proved that the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
implies a weaker bound of
on the size of the largest prime gaps.

By the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
, the expected number of primes between
and
is approximately
, and it is additionally known that for
almost all
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
intervals of this form the actual number of primes () is
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
to this expected number. Since this number is large for large
, this lends credence to Legendre's conjecture. It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally or based on the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
, but the lengths of the intervals for which this has been proven are longer than the intervals between consecutive squares, too long to prove Legendre's conjecture.
Partial results
It follows from a result by
Ingham that for all sufficiently large
, there is a prime between the consecutive ''
cubes
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
''
and
. Dudek proved that this holds for all
.
Dudek also proved that for
and any positive integer
, there is a prime between
and
. Mattner lowered this to
which was further reduced to
by Cully-Hugill.
Baker,
Harman, and
Pintz proved that there is a prime in the interval