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Let  p  be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if  p  ≡ ±3(mod  8), then the equation 8  x   +  p    y   =  z   2  has no positive integer solutions ( x ,  y ,  z ); (ii) if  p  ≡ 7(mod  8), then the equation has only the solutions ( p ,  x ,  y ,  z ) = (2  q   − 1, (1/3)( q  + 2), 2, 2  q   + 1), where  q  is an odd prime with  q  ≡ 1(mod  3); (iii) if  p  ≡ 1(mod  8) and  p  ≠ 17, then the equation has at most two positive integer solutions ( x ,  y ,  z ).

the scientific world journal/thescientificworldjournal

The Diophantine Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

The Diophantine Equation8x+py=z2