Generalization of a problem of Diophantus

Abstract

Greek mathematician Diophantus of Alexandria noted that the numbers 1 16 , 33 16 , 68 16 and 105 16 have the property that the product of any two of them when increased by 1 is a square of a rational number. Let n be an integer. We say the set of natural numbers {a1, a2, . . . , am} has the property of Diophantus of order n, in brief D(n), if for all i, j = 1, 2, . . . ,m, i 6= j, the following holds: ai · aj + n = bij , where bij is an integer. The first set of four natural numbers with property D(1) was found by French mathematician Pierre de Fermat (1601 1665). That set is {1, 3, 8, 120}. Davenport and Baker [3] show that a fifth integer r cannot be added to that set and maintain the same property unless r = 0. For the rational number r = 777480 8288641 the product of any two different members of that set increased by 1 is the square of a rational number (see [1]). In this paper we consider some problems of existence of sets of four natural numbers with property D(n), for any integer n. We prove that, for all e ∈ Z, there exist infinite numbers of sets of four natural numbers with property D(e2). Indeed, we show how a set {a, b} with property D(e2) can be extended to a set {a, b, c, d} with the same property, if a · b is not a perfect square. That construction is applied to the identities