On m-ary partition function congruences: A fresh look at a past problem

Øystein J. Rødseth; James A. Sellers

doi:10.1006/jnth.2000.2594

On m-ary partition function congruences: A fresh look at a past problem

Øystein J. Rødseth, James A. Sellers

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Abstract

Let b_m(n) denote the number of partitions of n into powers of m. Define σ_r=ε₂m²+ε₃m³+...+ε_rm^r, where ε_i=0 or 1 for each i. Moreover, let c_r=1 if m is odd, and c_r=2^r-1 if m is even. The main goal of this paper is to prove the congruence b_m(m^r+1n-σ_r-m)≡0 (modm^r/c_r). For σ_r=0, the existence of such a congruence was conjectured by R. F. Churchhouse some 30 years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after.

Original language	English (US)
Pages (from-to)	270-281
Number of pages	12
Journal	Journal of Number Theory
Volume	87
Issue number	2
DOIs	https://doi.org/10.1006/jnth.2000.2594
State	Published - Apr 2001
Externally published	Yes

Keywords

Congruences
Partitions

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10.1006/jnth.2000.2594

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title = "On m-ary partition function congruences: A fresh look at a past problem",

abstract = "Let bm(n) denote the number of partitions of n into powers of m. Define σr=ε2m2+ε3m3+...+εrmr, where εi=0 or 1 for each i. Moreover, let cr=1 if m is odd, and cr=2r-1 if m is even. The main goal of this paper is to prove the congruence bm(mr+1n-σr-m)≡0 (modmr/cr). For σr=0, the existence of such a congruence was conjectured by R. F. Churchhouse some 30 years ago, and its truth was proved by {\O}. J. R{\o}dseth, G. E. Andrews, and H. Gupta soon after.",

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T1 - On m-ary partition function congruences

T2 - A fresh look at a past problem

AU - Rødseth, Øystein J.

AU - Sellers, James A.

PY - 2001/4

Y1 - 2001/4

N2 - Let bm(n) denote the number of partitions of n into powers of m. Define σr=ε2m2+ε3m3+...+εrmr, where εi=0 or 1 for each i. Moreover, let cr=1 if m is odd, and cr=2r-1 if m is even. The main goal of this paper is to prove the congruence bm(mr+1n-σr-m)≡0 (modmr/cr). For σr=0, the existence of such a congruence was conjectured by R. F. Churchhouse some 30 years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after.

AB - Let bm(n) denote the number of partitions of n into powers of m. Define σr=ε2m2+ε3m3+...+εrmr, where εi=0 or 1 for each i. Moreover, let cr=1 if m is odd, and cr=2r-1 if m is even. The main goal of this paper is to prove the congruence bm(mr+1n-σr-m)≡0 (modmr/cr). For σr=0, the existence of such a congruence was conjectured by R. F. Churchhouse some 30 years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after.

KW - Congruences

KW - Partitions

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On m-ary partition function congruences: A fresh look at a past problem

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