On S-unit equations in two unknowns.
We consider systems of equations of the form and , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.
Let denote the ith prime. We conjecture that there are precisely 28 solutions to the equation in positive integers n and α₁,..., . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).
We consider an equation of the typeover the finite field . Carlitz obtained formulas for the number of solutions to this equation when and when and . In our earlier papers, we found formulas for the number of solutions when or or ; and when and is a power of modulo . In this paper, we obtain formulas for the number of solutions when , , or . For general case, we derive lower bounds for the number of solutions.
Assertions on the congruence f(x) + g(y) + c ≡ 0 (mod xy) made without proof by Mordell in his paper in Acta Math. 88 (1952) are either proved or disproved.
Let be an odd prime. By using the elementary methods we prove that: (1) if , the Diophantine equation has no positive integer solution except when or is of the form , where is an odd positive integer. (2) if , , then the Diophantine equation has no positive integer solution.