On certain combinatorial Diophantine equations and their connection to Pythagorean numbers

On Certain Combinatorial Diophantine Equations and Their Connection to Pythagorean Numbers

We call any (m+2)-tuple (n, r, r1, . . . , rm) satisfying this equation a binomial knapsack. The problem first came to the authors’ attention when considering a problem on symmetric functions, but the name is derived from the connection to knapsacktype problems. This article considers the simplest case of this problem. That is, we consider the problem of determining all 4-tuples (n, r, s, t) sa...

On Certain Diophantine Equations Related to Triangular and Tetrahedral Numbers

In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians NUMBER THEORY, Kraków, 14th20th september 2008.

Diophantine Equations and Class Numbers

The goals of this paper are to provide: (I ) sufficient conditions, based on the solvability of certain diophantine equations, for the non-triviality of the dass numbers of certain real quadratic fields; (2) sufficient conditions for the divisibility of the class numbers of certain imaginary quadratic fields by a given integer; and (3) necessary and sufficient conditions for an algebraic intege...

On a class of combinatorial diophantine equations

We give a combinatorial proof for a second order recurrence for the polynomials pn(x), where pn(k) counts the number of integer-coordinate lattice points x = (x1, . . . , xn) with ‖x‖ = ∑n i=1|xi| ≤ k. This is the main step to get finiteness results on the number of solutions of the diophantine equation pn(x) = pm(y) if n and m have different parity. The combinatorial approach also allows to ex...

On Certain Diophantine Equations in Rings and Fields.