View source for Newman's conjecture

{{short description|Unsolved problem in mathematics}}
{{Distinguish||text=the conjecture by Charles M. Newman about the [[de Bruijn–Newman constant]]}}{{Infobox mathematical statement
| name = Newman's conjecture
| image =
| caption =
| type = [[conjecture]]
| field = [[analytic number theory]]
| conjectured by = Morris Newman
| conjecture date =25 March 1960 
| open problem =Yes
| known cases =[[Prime power|Prime powers]] except [[powers of 2]] or powers of 3, plus selected other numbers (e.g. [[16 (number)|16]], [[40 (number)|40]], [[65 (number)|65]])
| first proof by =
| first proof date =
| implied by =
| equivalent to =
| generalizations =
| consequences =Erdős-Ivić conjecture
}}{{Unsolved|mathematics|Given arbitrary {{Mvar|m}}, {{Mvar|r}}, are there infinitely values of {{Mvar|n}} such that the partition function at {{Mvar|n}} is congruent to {{Mvar|r}} mod {{Mvar|m}}?}}
In [[mathematics]], specifically in [[number theory]], '''Newman's conjecture''' is a conjecture about the behavior of the [[Partition function (number theory)|partition function]] modulo any integer.  Specifically, it states that for any integers {{Mvar|m}} and {{Mvar|r}} such that <math>0\le r\le m-1</math>, the value of the partition function <math>p(n)</math> satisfies the congruence <math>p(n)\equiv r\pmod{m}</math> for infinitely many non-negative integers {{Mvar|n}}.  It was formulated by mathematician Morris Newman in 1960.<ref name=":1">{{Cite journal|last=Newman|first=Morris|date=1960|title=Periodicity Modulo m and Divisibility Properties of the Partition Function|journal=Transactions of the American Mathematical Society|volume=97|issue=2|pages=225–236|doi=10.2307/1993300|issn=0002-9947|jstor=1993300}}</ref> It is unsolved as of {{CURRENTYEAR}}.

==History==
Oddmund Kolberg was probably the first to prove a related result, namely that the [[Partition function (number theory)|partition function]] takes both even and odd values infinitely often. The proof employed was of elementary nature and easily accessible, and was proposed as an exercise by Newman in the [[American Mathematical Monthly]].<ref>{{Cite journal|last=Subbarao|first=M. V.|date=1966|title=Some Remarks on the Partition Function|journal=The American Mathematical Monthly|volume=73|issue=8|pages=851–854|doi=10.2307/2314179|issn=0002-9890|jstor=2314179}}</ref><ref>{{Cite journal|last=Kolberg|first=O.|date=1959-12-01|title=Note on the Parity of the Partition Functions.|journal=Mathematica Scandinavica|language=en|volume=7|pages=377–378|doi=10.7146/math.scand.a-10584|issn=1903-1807|doi-access=free}}</ref><ref>{{Cite journal|last=Newman|first=Morris|last2=van Lint|first2=J. H.|date=1962|title=4944|journal=The American Mathematical Monthly|volume=69|issue=2|pages=175|doi=10.2307/2312568|issn=0002-9890|jstor=2312568}}</ref>

1 year later, in 1960, Newman proposed the conjecture and proved the cases m=5 and 13 in his original paper,<ref name=":1" /> and m=65 two years later.<ref>{{Cite journal|last=Newman|first=Morris|date=March 1962|title=Congruences for the partition function to composite moduli|journal=Illinois Journal of Mathematics|language=EN|volume=6|issue=1|pages=59–63|doi=10.1215/ijm/1255631806|issn=0019-2082|doi-access=free}}</ref>

[[Ken Ono]], an American mathematician, made further advances by exhibiting sufficient conditions for the conjecture to hold for prime {{Mvar|m}}. He first showed that Newman's conjecture holds for prime {{Mvar|m}} if for each {{Mvar|r}} between 0 and {{Mvar|m-1}}, there exists a nonnegative integer {{Mvar|n}} such that the following holds:

*<math>24\mid mn+1 </math>
*<math>p \left (\frac{mn+1}{24}  \right ) \equiv r  \pmod{m}</math>

He used the result, together with a computer program, to prove the conjecture for all primes less than 1000 (except 3).<ref name=":0">{{Cite journal|last=Ono|first=Ken|date=2000|title=Distribution of the Partition Function Modulo m|journal=Annals of Mathematics|volume=151|issue=1|pages=293–307|doi=10.2307/121118|issn=0003-486X|jstor=121118|bibcode=2000math......8140O|arxiv=math/0008140}}</ref> Ahlgren expanded on his result to show that Ono's condition is, in fact, true for all composite numbers [[Coprime integers|coprime]] to 6.<ref>{{Cite journal|last=Ahlgren|first=Scott|date=2000-12-01|title=Distribution of the partition function modulo composite integers M|journal=Mathematische Annalen|language=en|volume=318|issue=4|pages=795–803|doi=10.1007/s002080000142|issn=1432-1807}}</ref>

Three years later, Ono showed that for every prime {{Mvar|m}} greater than 3, one of the following must hold:

* Newman's conjecture holds for {{Mvar|m}}, or
*<math>m\mid p(mn+k)</math> for all nonnegative integers {{Mvar|n}}, and <math>1\le k < 24,24k\equiv 1 \pmod{m}</math>.

Using computer technology, he proved the theorem for all primes less than 200,000 (except 3).<ref name=":2">{{Cite journal|last=Bruinier|first=Jan H.|last2=Ono|first2=Ken|date=2003-03-01|title=Coefficients of half-integral weight modular forms|journal=Journal of Number Theory|volume=99|issue=1|pages=164–179|doi=10.1016/S0022-314X(02)00061-6|issn=0022-314X|doi-access=free}}</ref>

Afterwards, Ahlgren and Boylan used Ono's criterion to extend Newman's conjecture to all primes except possibly 3.<ref>{{Cite journal|last=Ahlgren|first=Scott|last2=Boylan|first2=Matthew|date=2003-09-01|title=Arithmetic properties of the partition function|journal=Inventiones Mathematicae|language=en|volume=153|issue=3|pages=487–502|doi=10.1007/s00222-003-0295-6|issn=1432-1297|bibcode=2003InMat.153..487A}}</ref> 2 years afterwards, they extended their result to all prime powers except powers of 2 or 3.<ref>{{Cite journal|last=Ahlgren|first=Scott|last2=Boylan|first2=Matthew|date=2005-01-01|title=Coefficients of half-integral weight modular forms modulo ℓj|journal=Mathematische Annalen|language=en|volume=331|issue=1|pages=219–239|doi=10.1007/s00208-004-0555-9|issn=1432-1807}}</ref>

== Partial progress and solved cases ==
The weaker statement that <math>p(n)\equiv 0 \pmod{m}</math> has at least 1 solution has been proved for all {{Mvar|m}}. It was formerly known as the [[Erdős–Ivić conjecture]], named after mathematicians [[Paul Erdős]] and [[Aleksandar Ivić]]. It was settled by Ken Ono.<ref name=":0" />

== References ==
{{Reflist}}

[[Category:Unsolved problems in number theory]]
[[Category:Analytic number theory]]