C. L. Stewart

On the number of solutions of polynomial congruences and Thue equations

has only finitely many solutions in integers x and yv. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of Nagell [30], Ore [32], Sandor [33], and Huxley [19]. In fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is, in general, best possible. In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gy6ry, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.

On shifted products which are powers

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai’s are rational numbers. It is conjectured that there do not exist five positive integers such that their pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is the set of positive integers of the form x with x and k positive integers and k > 1. How large can a set of positive integers A be if aa + 1 is in V whenever a and a are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

Some Ramanujan-Nagell equations with many solutions

If we fix y as 1 in (1) we obtain a Ramanujan-Nagell equation. In [4] Erdös, Stewart and Tijdeman proved that the exponential dependence on s in estimates (2) and (3) is not far from the truth by giving examples of Ramanujan-Nagell equations with many solutions. Let ε be a positive number, let 2 = p1, p2, . . . be the sequence of prime numbers and let n be an integer with n ≥ 2. They proved that there exists a number s0, which is effectively computable in terms of ε and n, such that if s is an integer with

On sums which are powers

The cardinality of sets A is estimated under the conditions that every element of the sum set A+A is a power resp. powerful number (n is said to be powerful if pn implies p2n). Subset sums with these properties are also studied.