Given m, n ≥ 2, we prove that, for sufficiently large y, the sum 1 n +···+ y n is not a product of m consecutive integers. We also prove that for m ≠ n we have 1 m +···+ x m ≠ 1 n +···+ y n , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions.

In this paper we give a new bound for the solutions x of the title equation, provided that k greater than or equal to 8. This bound is polynomial in d. Moreover, under the same condition, a similar bound for the number of solutions in (x, k, y, l) is given.

A linear recursive sequence G of order k is defined by the integer initial terms G(0),G(1),...,G(k-1), integer constansts A(1), A(2), ... A(k) and by the recursion G(n) = A(1)G(n-1) +...+f A(k)G(n-k) for k less than or equal to n. In the case k = 2, it is known that there are generally only finitely many perfect powers in the sequence. T.N. Shorey and C.L. Stewart showed that if the sequence G has a simple dominating characteristic zero, then does not contain q-th powers if q is large enough. P. Kiss proved that under some conditions the equation G(n)G(x) = w(q) in positive integers x, w, q has no solution with x > n and q > q(0)(n). We show that if a product G(x)H(y), where G and H are not necessarily identical recurrence sequences and x, y are not too far from each other, is a q-th power then q is less than a bound which is effectively computable.

In the paper a new method is given to derive a reasonable upper bound for the number of solutions of the generalized Ramanujan-Nagell equation.

In this paper we give a new, generalized version of a result of Brindza, Evertse and Gyory, concerning superelliptic equations.

B. Brindza and A. Pint´ ´er (Debrecen)To the memory of Paul Erd˝osLet f(X) and g(Y ) be polynomials with integral coefficients in the singleindependent variables X and Y . The diophantine problem f(x) = g(y) isstrongly related to the absolute irreducibility and the genus of f(X)−g(Y )as pointed out by Davenport, Lewis and Schinzel [DLS]:Theorem A. Let f(X) be of degree n > 1 and g(Y ) of degreem > 1. Let D(λ) = disc(f(x) + λ) and E(λ) = disc(g(y) + λ). Supposethere are at least [n/2] distinct roots of D(λ) = 0 for which E(λ) 6= 0 . Thenf(X)−g(Y ) is irreducible over the complex field. Further, the genus of theequation f(x) − g(y) = 0 is strictly positive except possibly when m = 2 orm = n = 3. Apart from these possible exceptions, the equation has at mosta finite number of integral solutions.The purpose of this note is to handle some special cases. For an integerk > 1 we setf

The covering properties of Artinian rings which depend on their additive structure only, are investigated.

Let D(An) denote the discriminant of the characteristic polynomial of thenth power of the matrixA. In this paper the polynomial values of D(An) are investigated and it is pointed out that the ring generated by the spectrum ofAplays an important role.

As it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant -n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.

As it was pointed out by Lang [4, p. 245] and others, certain finiteness results for diophantine equations over algebraic number fields can be extended, by using deep algebraic geometrical arguments, to rather general cases when the ground domain of unknowns is a finitely generated field or a finitely generated subring of it. The purpose of this paper is to establish a surprisingly elementary method, through a concrete equation, to obtain these kind of general results. Let f(X,Y ) and g(X,Y ) be binary forms (homogeneous polynomials in two variables) with complex coefficients of degree m and n, respectively. The binary form fg splits into linear factors (over C) and in the sequel, we suppose that the linear factors are pairwise non-proportional. Let K be a finitely generated subfield of C. Then K can be written in the form Q(z1, . . . , zq, u), where z1, . . . , zq is a transcendence basis of K and we may assume without loss of generality that the element u is integral over the polynomial ring Z[z1, . . . , zq]. Theorem. If n ≥ 1 and m− n ≥ 5 then the equation