ON THE DIOPHANTINE EQUATION (8n)x+(15n)y=(17n)z

Let a , b , c be relatively prime positive integers such that a 2 + b 2 = c 2 . Half a century ago, Jeśmanowicz [‘Several remarks on Pythagorean numbers’, Wiadom. Mat. 1 (1955/56), 196–202] conjectured that for any given positive integer n the only solution of ( an ) x +( bn ) y =( cn ) z in positive integers is ( x , y , z )=(2,2,2). In this paper, we show that (8 n ) x +(15 n ) y =(17 n ) z has no solution in positive integers other than ( x ,...