Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician 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 integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. He obtained his B.Sc from
Middle East Technical University in
Ankara, Turkey
Ankara ( , ; ), historically known as Ancyra and Angora, is the capital of Turkey. Located in the central part of Anatolia, the city has a population of 5.1 million in its urban center and over 5.7 million in Ankara Province, mak ...
and his PhD from the
University of Toronto
The University of Toronto (UToronto or U of T) is a public university, public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park (Toronto), Queen's Park. It was founded by royal charter in 1827 ...
in 1990. His advisor was
John Friedlander. He is currently a faculty member at
Boğaziçi University
Boğaziçi University ( tr, Boğaziçi Üniversitesi), also known as Bosphorus University, is a major research university in Istanbul, Turkey. Its main campus is located on the European side of the Bosphorus strait. It has six faculties and tw ...
in
Istanbul, Turkey
)
, postal_code_type = Postal code
, postal_code = 34000 to 34990
, area_code = +90 212 (European side) +90 216 (Asian side)
, registration_plate = 34
, blank_name_sec2 = GeoTLD
, blank_in ...
.
In 2005(), with
Dan Goldston and
János Pintz
János Pintz (born 20 December 1950 in Budapest) 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. In 2014, he receiv ...
, 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 ''p''
''n'' denotes 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 ways ...
. In other words, for every ''c'' > 0, there exist infinitely many pairs of consecutive primes ''p''
''n'' and ''p''
''n''+1 which are closer to each other than the average distance between consecutive primes by a factor of ''c'', i.e., ''p''
''n''+1 − ''p''
''n'' < ''c'' log ''p''
''n''.
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
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, who ...
, then they can also show that primes within 16 of each other occur infinitely often, which is related to the
twin prime conjecture
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
.
See also
*
Landau's problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau ...
References
*
External links
*
20th-century Turkish mathematicians
21st-century Turkish mathematicians
1961 births
Living people
Number theorists
Institute for Advanced Study visiting scholars
University of Toronto alumni
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