Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 .Jesmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an) x + (bn) v = (en)* in positive integers is x -y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > l , c = b + l and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples

Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an) x + (bn) y = (cn) z in positive integers is x = y = z = 2. If n = 1, then, equivalently, the equation (u 2 −v 2) x +(2uv) y = (u 2 +v 2) z , for integers u > v > 0, has only the solution x = y = z = 2. We prove that this is the case when one of u, v has no prime factor of the form 4l + 1 and certain congruence and inequality conditions on u, v are satisfied.

Let q be an odd prime. Let c > 1 and t be positive integers such that q t þ 1 ¼ 2c 2 . Using elementary method and a result due to Ljunggren concerning the Diophantine equation x n À1 xÀ1 ¼ y 2 , we show that the Diophantine equation x 2 þ q m ¼ c 2n has the only positive integer solution ðx; m; nÞ ¼ ðc 2 À 1; t; 2Þ. As applications of this result some new results on the Diophantine equation x 2 þ q m ¼ c n and the Diophantine equation x 2 þ ð2c À 1Þ m ¼ c n are obtained. In particular, we prove that Terai's conjecture is true for c ¼ 12; 24. Combining this result with Terai's results we conclude that Terai's conjecture is true for 2 c 30.

No abstract