Find-integer-x-y-such-that-2-x-y-2-615-

Question Number 80053 by mr W last updated on 30/Jan/20

$${Find}\:{integer}\:{x},\:{y}\:{such}\:{that} \\ $$$$\mathrm{2}^{{x}} −{y}^{\mathrm{2}} =\mathrm{615} \\ $$

Commented by john santu last updated on 30/Jan/20

((√2^x ))^2 −(y)^2 =615 ((√(2^x ))−y)((√2^x ) +y)=615 = 123×5 (√2^x ) +y=123 ∧ (√2^x ) −y=5 (√2^x ) = 64=2^6 ⇒x=12 , y = 59

$$\left(\sqrt{\mathrm{2}^{\mathrm{x}} }\right)^{\mathrm{2}} −\left(\mathrm{y}\right)^{\mathrm{2}} =\mathrm{615} \\ $$$$\left(\sqrt{\mathrm{2}^{\mathrm{x}} \:}−\mathrm{y}\right)\left(\sqrt{\mathrm{2}^{\mathrm{x}} }\:+\mathrm{y}\right)=\mathrm{615}\:=\:\mathrm{123}×\mathrm{5} \\ $$$$\sqrt{\mathrm{2}^{\mathrm{x}} }\:+\mathrm{y}=\mathrm{123}\:\wedge\:\sqrt{\mathrm{2}^{\mathrm{x}} }\:−\mathrm{y}=\mathrm{5} \\ $$$$\sqrt{\mathrm{2}^{\mathrm{x}} }\:=\:\mathrm{64}=\mathrm{2}^{\mathrm{6}} \:\Rightarrow\mathrm{x}=\mathrm{12}\:,\:\mathrm{y}\:=\:\mathrm{59} \\ $$

Commented by mr W last updated on 30/Jan/20