{}1. Introduction. In 1956, Sierpifiski [7] showed that the equation 3x q-4 5 z has the only positive integral solution (x, y, z)= (2, 2, 2).And it is conjectured that if a, b, c are a Pythagorean triplet, i.e. positive inb z tegers satisfying a -t---c, then the Diophantine equation aX+ b-c has the only positive integral solution (x, y, z)--(2, 2, 2). It has been verified that this conjecture holds for many other Pythagorean triplets (el. Sierpiflski[8], Jemanowicz [3], Lu[4], Takakuwa and Asaeda [9], [10], Takakuwa [11]. See also Terai [12]).As an analogy of this conjecture, we consider the following:Conjecture. If a, b, c, p, q, r are fixed positive integers satisfying a b c with p, q, r >_ 2, then the Diophantine equation(1) a + b c has the only positive integral solution (x, !t, z)= (p, q, r).We note that Scott [6] proved that if a and b are relatively prime integers greater than one, and if c is prime, then the equation ax -k bc has