Viet Kh. Nguyen
CombinatoricsParity (mathematics)Prime (order theory)ChemistryTree of primitive Pythagorean triplesDiophantine equationConnection (algebraic framework)ConjectureMathematicsIntegerPythagorean triple
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Let (a, b, c)be a primitive Pythagorean triple parameterized as a=u^2-v^2,\ b=2uv,\ c=u^2+v^2\ where u>v>0are co-prime and not of the same parity. In 1956, L. Je{\'s}manowicz conjectured that for any positive integer n the Diophantine equation (an)^x+(bn)^y=(cn)^zhas only the positive integer solution (x,y,z)=(2,2,2) In this connection we call a positive integer solution (x,y,z)\ne (2,2,2)with n>1exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of...