In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, Rosser's theorem states that the ''n''th
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only way ...
is greater than
, where
is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
function. It was published by
J. Barkley Rosser in 1939.
Its full statement is:
Let ''p''
''n'' be the ''n''th
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only way ...
. Then for ''n'' ≥ 1
:
In 1999,
Pierre Dusart
Pierre Dusart is a French mathematician at the Université de Limoges who specializes in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the i ...
proved a tighter lower bound:
[{{cite journal, authorlink=Pierre Dusart, last=Dusart, first=Pierre, title=The {{mvar, kth prime is greater than {{math, ''k''(log ''k'' + log log ''k''−1) for {{math, ''k'' ≥ 2, journal=]Mathematics of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than f ...
, volume=68, issue=225, year=1999, pages=411–415, mr=1620223, doi=10.1090/S0025-5718-99-01037-6, doi-access=free
:
See also
*
Prime number theorem
References
External links
Rosser's theoremarticle on Wolfram Mathworld.
Theorems about prime numbers
de:John Barkley Rosser#Satz von Rosser