János Pintz (born 20 December 1950 in
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population ...
) is a
Hungarian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
working in
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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. He is a fellow of the
Rényi Mathematical Institute and is also a member of the
Hungarian Academy of Sciences. In 2014, he received the
Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number ...
.
Mathematical results
Pintz is best known for proving in 2005 (with
Daniel Goldston
Daniel Alan Goldston (born January 4, 1954 in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University.
Early life and education
Daniel Alan Goldst ...
and
Cem Yıldırım
Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician who specializes in number theory. He obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His advisor ...
) that
::
where
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 ε > 0, there exist infinitely many pairs of consecutive primes ''p''
''n'' and ''p''
''n''+1 that are closer to each other than the average distance between consecutive primes by a factor of ε, i.e., ''p''
''n''+1 − ''p''
''n'' < ε log ''p''
''n''. This result was originally reported in 2003 by
Daniel Goldston
Daniel Alan Goldston (born January 4, 1954 in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University.
Early life and education
Daniel Alan Goldst ...
and
Cem Yıldırım
Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician who specializes in number theory. He obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His advisor ...
but was later retracted.
Pintz joined the team and completed the proof in 2005. Later they improved this to showing that ''p''
''n''+1 − ''p''
''n'' < ε(log log ''n'')
2 occurs infinitely often. Further, if one assumes 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 st ...
, then one can also show that primes within 16 of each other occur infinitely often, which is nearly 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 ...
.
Additionally,
* With
János Komlós and
Endre Szemerédi
Endre Szemerédi (; born August 21, 1940) is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science ...
he disproved the
Heilbronn conjecture.
* With
Iwaniec he proved that for sufficiently large ''n'' there is a prime between ''n'' and ''n'' + ''n''
23/42.
* Pintz gave an effective upper bound for the first number for which the
Mertens conjecture
In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 ...
fails.
* He gave an O(''x''
2/3) upper bound for the number of those numbers that are less than ''x'' and not the sum of two primes.
* With
Imre Z. Ruzsa he improved a result of
Linnik by showing that every sufficiently large even number is the sum of two primes and at most 8 powers of 2.
* Goldston, S. W. Graham, Pintz, and Yıldırım proved that the difference between numbers which are products of exactly 2 primes is infinitely often at most 6.
[D. Goldston, S. W. Graham, J. Pintz, C. Yıldırım: Small gaps between products of two primes, Proc. Lond. Math. Soc., 98(2007) 741–774.]
See also
*
Prime gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the
''n''-th prime numbers, i.e.
:g_n = p_ - p_n.\
W ...
*
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 ...
*
Fazekas Mihály Gimnázium
*
Maier's theorem In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer.
The theorem states that if π is the prime-counting function and λ is greater ...
References
External links
János Pintz's pageat the
Alfréd Rényi Institute of Mathematics
*
{{DEFAULTSORT:Pintz, Janos
Number theorists
Institute for Advanced Study visiting scholars
Members of the Hungarian Academy of Sciences
Eötvös Loránd University alumni
Mathematicians from Budapest
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Living people
1950 births