Cem Yildirim
Encyclopedia
Cem Yalçın Yıldırım is a Turkish
mathematician
who specializes in number theory
. He obtained his PhD from the University of Toronto
in 1990. His advisor was John Friedlander
. He is currently a faculty member at Boğaziçi University
in Istanbul, Turkey.
In 2005, with Dan Goldston and János Pintz
, he proved, that for any positive number ε there exist primes p and p′ such that the difference between p and p′ is smaller than ε log p.
Formally;
where pn denotes the nth prime number
. In other words, for every c>0, there exist infinitely many pairs of consecutive primes pn and pn+1 which are closer to each other than the average distance between consecutive primes by a factor of c, i.e., pn+1-pn<c log pn.
This result was originally reported in 2003 by Dan Goldston and Cem Yıldırım but was later retracted. Then Janos Pintz joined the team and they completed the proof in 2005.
In fact, if they assume the Elliott-Halberstam conjecture
, then they can also show that primes within 16 of each other occur infinitely often, which is related to the twin prime conjecture
.
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who specializes in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. He obtained his PhD from the University of Toronto
University of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...
in 1990. His advisor was 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...
. He is currently a faculty member at Boğaziçi University
Bogaziçi University
Boğaziçi University is a public university located on the European side of the Bosphorus strait in Istanbul, Turkey. It has five faculties and two schools offering undergraduate degrees, and six institutes offering graduate degrees...
in Istanbul, Turkey.
In 2005, with Dan Goldston and János Pintz
János Pintz
János Pintz is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences.-Mathematical results:...
, he proved, that for any positive number ε there exist primes p and p′ such that the difference between p and p′ is smaller than ε log p.
Formally;
where pn denotes the nth 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...
. In other words, for every c>0, there exist infinitely many pairs of consecutive primes pn and pn+1 which are closer to each other than the average distance between consecutive primes by a factor of c, i.e., pn+1-pn<c log pn.
This result was originally reported in 2003 by Dan Goldston and Cem Yıldırım but was later retracted. Then Janos Pintz joined the team and they completed the proof in 2005.
In fact, if they assume the Elliott-Halberstam conjecture
Elliott-Halberstam conjecture
In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam....
, then they can also show that primes within 16 of each other occur infinitely often, which is related to the twin prime conjecture
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
.