Equal values of certain partition functions via Diophantine equations

A New Method for Certain Diophantine Equations

1. Introduction. The original purpose of the research described in this article was to obtain results about Diophantine problems on rational surfaces — that is, surfaces defined over a field k which are birationally equivalent to P 2 over the algebraic closure ¯ k. (Throughout this article, K and k will always denote algebraic number fields, with respective rings of integers O and o. Except in ...

Diophantine Equations Involving Arithmetic Functions of Factorials

DIOPHANTINE EQUATIONS INVOLVING ARITHMETIC FUNCTIONS OF FACTORIALS Daniel M. Baczkowski We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation:

On Powers of 2 Dividing the Values of Certain Plane Partition Functions

We consider two families of plane partitions: totally symmetric self-complementary plane partitions (TSSCPPs) and cyclically symmetric transpose complement plane partitions (CSTCPPs). If T (n) and C(n) are the numbers of such plane partitions in a 2n× 2n× 2n box, then ord2(T (n)) = ord2(C(n)) for all n ≥ 1. We also discuss various consequences, along with other results on ord2(T (n)). 2000 Math...

Parity Results for Certain Partition Functions

Blecksmith, Brillhart and Gerst proved four congruences modulo 2 involving partition generating functions of the following sort. 1 . Y n∈S (1 − q) ≡ 1 + X n≥1 q 2 + X n≥1 q 2 (mod 2) where S = {n > 0 : n ≡ ±(1, 2, 3, 4) or 6 (mod 12)}. We give simple and uniform proofs of their congruences and of several others of the same sort. Each of these congruences yields a theorem on partitions. Thus the...

On Certain Diophantine Equations in Rings and Fields.