Legendre's conjecture
Encyclopedia
Legendre's conjecture, proposed by Adrien-Marie Legendre
, states that there is a prime number
between n2 and (n + 1)2 for every positive integer n. The conjecture
is one of Landau's problems
(1912) and unproven .
The prime number theorem
suggests the actual number of primes between n2 and (n + 1)2 is about n/log(n), i.e. about as many as the number of primes less than or equal to n.
If Legendre's conjecture is true, the gap between any two successive primes would be
. In fact the conjecture follows from Andrica's conjecture. Harald Cramér
conjectured that the gap is always much smaller,
; if Cramér's conjecture is true, Legendre's conjecture would follow. Cramér also proved that the Riemann hypothesis
implies a weaker bound of
on the size of the largest prime gaps. Legendre's conjecture implies that at least one prime can be found in every revolution of the Ulam spiral
.
Legendre's conjecture follows from Opperman's conjecture.
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
, states that there is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
between n2 and (n + 1)2 for every positive integer n. The conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
is one of Landau's problems
Landau's problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of science" and are now known as Landau's problems...
(1912) and unproven .
The prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
suggests the actual number of primes between n2 and (n + 1)2 is about n/log(n), i.e. about as many as the number of primes less than or equal to n.
If Legendre's conjecture is true, the gap between any two successive primes would be
. In fact the conjecture follows from Andrica's conjecture. Harald CramérHarald Cramér
Harald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. He was once described by John Kingman as "one of the giants of statistical theory".-Early life:Harald Cramér was born in Stockholm, Sweden on September...
conjectured that the gap is always much smaller,
; if Cramér's conjecture is true, Legendre's conjecture would follow. Cramér also proved that the Riemann hypothesisRiemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
implies a weaker bound of
on the size of the largest prime gaps. Legendre's conjecture implies that at least one prime can be found in every revolution of the Ulam spiralUlam spiral
The Ulam spiral, or prime spiral is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes...
.
Legendre's conjecture follows from Opperman's conjecture.