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Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions ( x ,  y ,  z ) of the system of Diophantine equations  x   2  − 6 y   2  = −5 and  x  = 2 z   2  − 1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations  x   2  − 6 y   2  = −5 and  x  =  az   2  −  b  for each pair of integral parameters  a ,  b . The proof utilizes algebraic number theory and  p -adic analysis which successfully avoid discussing the class number and factoring the ideals.

the scientific world journal/thescientificworldjournal

On the System of Diophantine Equations<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>x</mml:mi><mml:mo>=</…

On the System of Diophantine Equationsx2-6y2=-5andx=