Some diophantine equations

Abstract

For positive integers x, y, the equation x4 + (n2-2)y - z always has the trivial solution x - y. In Chapter 1, we discuss the conditions under which the above equation cannot have any non-trivial solutions in positive integers. We also prove that if the above equation has no non-trivial solutions, then the 1st, 3rd, (n+1)th, (n+3)th terms of an arithmetical progression cannot each be square. In Chapter 2, we prove that any set of positive integers, with the property that the product of any two integers increased by 2 is a perfect square, can have at most three elements. We also prove that there exist infinitely many sets of four positive integers with the property that the product of any two increased by 1 is a perfect square. Although in general we could not-prove that a fifth integer cannot be added to these sets without altering the property, we prove it for a particular set {2, 4, 12, 420}. We also give an algebraic formula to find the fourth member of the set, if any three members are given. In Chapter 3, we prove that the only positive integer solutions of the equation (x(x - 1))2 = 3y(y - l) are (x, y) = (1, 1) (3, 4). In Chapter 4, we prove that the only positive integer solution of the equation 3y(y + 1) = x(x+1)(x+2)(x+3) is (x,y) = (12,104).