Bombieri–Friedlander–Iwaniec theorem
Encyclopedia
In analytic number theory
, an advanced branch of mathematics
, the Friedlander–Iwaniec theorem asserts that there are infinitely many prime number
s of the form
. The first few such primes are
The theorem was proved in 1997 by John Friedlander
and Henryk Iwaniec
, using sieve
techniques first developed by Enrico Bombieri
.
Iwaniec was awarded the 2001 Ostrowski Prize
in part for his contributions to this work.
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
, an advanced branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Friedlander–Iwaniec theorem asserts that there are infinitely many 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...
s of the form
. The first few such primes are- 2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, … .
The theorem was proved in 1997 by John Friedlander
John Friedlander
John Benjamin Friedlander is a Canadian mathematician specializing in analytic number theory. He received his B.Sc. from the University of Toronto in 1965, an M.A. from the University of Waterloo in 1966, and a Ph.D. from Pennsylvania State University in 1972. He was a lecturer at M.I.T...
and Henryk Iwaniec
Henryk Iwaniec
Henryk Iwaniec is a Polish American mathematician, and since 1987 a professor at Rutgers University. He was awarded the fourteenth Frank Nelson Cole Prize in Number Theory in 2002. He received the Leroy P. Steele Prize for Mathematical Exposition in 2011.-Background and education:Iwaniec studied...
, using sieve
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...
techniques first developed by Enrico Bombieri
Enrico Bombieri
Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...
.
Iwaniec was awarded the 2001 Ostrowski Prize
Ostrowski Prize
The Ostrowski Prize is a mathematics award given every other year by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands...
in part for his contributions to this work.