Equal values of standard counting polynomials

Abstract

The following discrete geometrical question provides a background for some classical diophantine problems. For given positive integers m, n, can an mdimensional and an n-dimensional unit cube, simplex, pyramid or octahedron contain equally many integral points? Apart from some trivial cases, the question leads to 9 families of diophantine equations, see Table 1. In this paper we give a brief survey of known results on these equations, and prove some new theorems concerning the solutions.

Polynomial values of sums of products of consecutive integers

“…The polynomial values of combinatorial polynomials have a vast literature. Here we only mention the papers [2,5,8,10], and the references given there. In this paper, we consider the problem of describing the polynomial values of a family of polynomials related to the sums of products of consecutive integers.…”

Polynomial values of products of terms from an arithmetic progression

“…We mention that there are many results in the literature which are related in the sense that they concern equal values or polynomial values of terms of families of combinatorial polynomials. We cannot survey the extremely huge literature, we only refer to the papers [1,4,8,[15][16][17]24] and the references there. The structure of the paper is the following.…”

On a conjecture of Schäffer concerning the equation1k+⋯+xk=ynHajdu1 2015Journal of Number Theory1301301View full textAdd to dashboardCite

“…Further, under certain assumptions Pintér [16] proved that for the nontrivial solutions we have n < ck log(2k), where c is an effectively computable absolute constant. For more results concerning (1) and its various generalizations, we refer once again to the book [20] and the papers [2, 6,7] and the references therein.…”

“…For details and more history we refer to the book [20] and the papers [2, 6,7] and the references given there. As it is well-known, in the case when (k, n) belongs to the set (2) {(1, 2), (3, 2), (3, 4), (5, 2)}…”