Legendre's constant
Encyclopedia
Legendre's constant is a mathematical constant
occurring in a formula conjectured by Adrien-Marie Legendre
to capture the asymptotic behavior of the prime-counting function
. Its value is now known to be exactly 1.
Examination of available numerical evidence for known primes
led Legendre to suspect that
satisfies:
where B is Legendre's constant. He guessed B to be about 1.08366, but regardless of its exact value, the existence of B implies the prime number theorem
.
Later Carl Friedrich Gauss
also examined the numerical evidence and concluded that the limit might be lower.
Charles Jean de la Vallée-Poussin
, who proved the prime number theorem (independently from Jacques Hadamard
), finally showed that B is 1.
Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
occurring in a formula conjectured by Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
to capture the asymptotic behavior of the prime-counting function
. Its value is now known to be exactly 1.Examination of available numerical evidence for known primes
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...
led Legendre to suspect that
satisfies:where B is Legendre's constant. He guessed B to be about 1.08366, but regardless of its exact value, the existence of B implies 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....
.
Later Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
also examined the numerical evidence and concluded that the limit might be lower.
Charles Jean de la Vallée-Poussin
Charles Jean de la Vallée-Poussin
Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...
, who proved the prime number theorem (independently from Jacques Hadamard
Jacques Hadamard
Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
), finally showed that B is 1.
Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.