Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple with $$a^2+b^2=c^2$$ a 2 + b 2 = c 2 . A positive integer solution (x, y, z) of the equation $$(an)^x+(bn)^y=(cn)^z$$ ( a n ) x + ( b n ) y = ( c n ) z is called exceptional if $$(x,y,z)\ne (2,2,2)$$ ( x , y , z ) ≠ ( 2 , 2 , 2 ) . Sixty years ago, L. Jeśmanowicz conjectured that, for any n, the equation has no exceptional solutions. This problem is not resolved as yet. In this paper, using the Baker method, we prove that if $$n>1$$ n > 1 , $$b+1=c$$ b + 1 = c and $$c>500000$$ c > 500000 , then the equation has no exceptional solutions (x, y, z) with $$y>z>x$$ y > z > x .
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas – Springer Journals
Published: Feb 28, 2017
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