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Let  b  and  c  be fixed coprime odd positive integers with min{ b ,  c } > 1. In this paper, a classification of all positive integer solutions ( x ,  y ,  z ) of the equation 2  x   +  b    y   =  c    z   is given. Further, by an elementary approach, we prove that if  c  =  b  + 2, then the equation has only the positive integer solution ( x ,  y ,  z ) = (1,1, 1), except for ( b ,  x ,  y ,  z ) = (89,13,1, 2) and (2  r   − 1,  r  + 2,2, 2), where  r  is a positive integer with  r  ≥ 2.

the scientific world journal/thescientificworldjournal

The Exponential Diophantine Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mn>2</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>b</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>c</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msup></mml:math>

The Exponential Diophantine Equation2x+by=cz