Legendre's conjecture explained
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between
and
for every
positive integer
. The
conjecture is one of
Landau's problems (1912) on prime numbers, and is one of many
open problems 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. It is one of a family of results and conjectures related to
prime gaps, that is, to the spacing between prime numbers. Others include
Bertrand's postulate, 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 also proved that the
Riemann hypothesis implies a weaker bound of
on the size of the largest prime gaps.
[1] By the prime number theorem, the expected number of primes between
and
is approximately
, and it is additionally known that for
almost all intervals of this form the actual number of primes is
asymptotic to this expected number. Since this number is large for large
, this lends credence to Legendre's conjecture.
[2] 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, 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
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
[3] which was further reduced to
by Cully-Hugill.
[4] Baker, Harman, and Pintz proved that there is a prime in the interval
for all large
.
A table of maximal prime gaps shows that the conjecture holds to at least
, meaning
.
[5] Notes and References
- .
- see p. 52, "It appears doubtful that this super-abundance of primes can be clustered insuch a way so as to avoid appearing at least once between consecutive squares."
- Mattner . Caitlin . 2017 . Prime Numbers in Short Intervals . BSc . Australian National University . 10.25911/5d9efba535a3e. en .
- Cully-Hugill . Michaela . 2023-06-01 . Primes between consecutive powers . Journal of Number Theory . 247 . 100–117 . 10.1016/j.jnt.2022.12.002 . 0022-314X. 2107.14468 .
- .