Newman's conjecture
| Type | conjecture |
|---|---|
| Field | analytic number theory |
| Conjectured by | Morris Newman |
| Conjectured in | 25 March 1960 |
| Open problem | Yes |
| Known cases | Prime powers except powers of 2 or powers of 3, plus selected other numbers (e.g. 16, 40, 65) |
| Consequences | Erdős-Ivić conjecture |
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Unsolved problem in mathematics: Given arbitrary m, r, are there infinitely values of n such that the partition function at n is congruent to r mod m? (more unsolved problems in mathematics)
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In mathematics, specifically in number theory, Newman's conjecture is a conjecture about the behavior of the partition function modulo any integer. Specifically, it states that for any integers m and r such that [math]\displaystyle{ 0\le r\le m-1 }[/math], the value of the partition function [math]\displaystyle{ p(n) }[/math] satisfies the congruence [math]\displaystyle{ p(n)\equiv r\pmod{m} }[/math] for infinitely many non-negative integers n. It was formulated by mathematician Morris Newman in 1960.[1] It is unsolved as of 2020.
History
Oddmund Kolberg was probably the first to prove a related result, namely that the 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.[2][3][4]
1 year later, in 1960, Newman proposed the conjecture and proved the cases m=5 and 13 in his original paper,[1] and m=65 two years later.[5]
Ken Ono, an American mathematician, made further advances by exhibiting sufficient conditions for the conjecture to hold for prime m. He first showed that Newman's conjecture holds for prime m if for each r between 0 and m-1, there exists a nonnegative integer n such that the following holds:
- [math]\displaystyle{ 24\mid mn+1 }[/math]
- [math]\displaystyle{ 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).[6] Ahlgren expanded on his result to show that Ono's condition is, in fact, true for all composite numbers coprime to 6.[7]
Three years later, Ono showed that for every prime m greater than 3, one of the following must hold:
- Newman's conjecture holds for m, or
- [math]\displaystyle{ m\mid p(mn+k) }[/math] for all nonnegative integers n, and [math]\displaystyle{ 1\le k \lt 24,24k\equiv 1 \pmod{m} }[/math].
Using computer technology, he proved the theorem for all primes less than 200,000 (except 3).[8]
Afterwards, Ahlgren and Boylan used Ono's criterion to extend Newman's conjecture to all primes except possibly 3.[9] 2 years afterwards, they extended their result to all prime powers except powers of 2 or 3.[10]
Partial progress and solved cases
The weaker statement that [math]\displaystyle{ p(n)\equiv 0 \pmod{m} }[/math] has at least 1 solution has been proved for all m. It was formerly known as the Erdős–Ivić conjecture, named after mathematicians Paul Erdős and Aleksandar Ivić (de). It was settled by Ken Ono.[6]
References
- ↑ 1.0 1.1 Newman, Morris (1960). "Periodicity Modulo m and Divisibility Properties of the Partition Function". Transactions of the American Mathematical Society 97 (2): 225–236. doi:10.2307/1993300. ISSN 0002-9947.
- ↑ Subbarao, M. V. (1966). "Some Remarks on the Partition Function". The American Mathematical Monthly 73 (8): 851–854. doi:10.2307/2314179. ISSN 0002-9890.
- ↑ Kolberg, O. (1959-12-01). "Note on the Parity of the Partition Functions." (in en). Mathematica Scandinavica 7: 377–378. doi:10.7146/math.scand.a-10584. ISSN 1903-1807.
- ↑ Newman, Morris; van Lint, J. H. (1962). "4944". The American Mathematical Monthly 69 (2): 175. doi:10.2307/2312568. ISSN 0002-9890.
- ↑ Newman, Morris (March 1962). "Congruences for the partition function to composite moduli" (in EN). Illinois Journal of Mathematics 6 (1): 59–63. doi:10.1215/ijm/1255631806. ISSN 0019-2082.
- ↑ 6.0 6.1 Ono, Ken (2000). "Distribution of the Partition Function Modulo m". Annals of Mathematics 151 (1): 293–307. doi:10.2307/121118. ISSN 0003-486X. Bibcode: 2000math......8140O.
- ↑ Ahlgren, Scott (2000-12-01). "Distribution of the partition function modulo composite integers M" (in en). Mathematische Annalen 318 (4): 795–803. doi:10.1007/s002080000142. ISSN 1432-1807.
- ↑ Bruinier, Jan H.; Ono, Ken (2003-03-01). "Coefficients of half-integral weight modular forms". Journal of Number Theory 99 (1): 164–179. doi:10.1016/S0022-314X(02)00061-6. ISSN 0022-314X.
- ↑ Ahlgren, Scott; Boylan, Matthew (2003-09-01). "Arithmetic properties of the partition function" (in en). Inventiones Mathematicae 153 (3): 487–502. doi:10.1007/s00222-003-0295-6. ISSN 1432-1297. Bibcode: 2003InMat.153..487A.
- ↑ Ahlgren, Scott; Boylan, Matthew (2005-01-01). "Coefficients of half-integral weight modular forms modulo ℓj" (in en). Mathematische Annalen 331 (1): 219–239. doi:10.1007/s00208-004-0555-9. ISSN 1432-1807.
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