MathLabs

Problem 4

Determine all pairs (x,y)(x,y) of integers such that 1+2x+22x+1=y21+2^{x}+2^{2x+1}=y^{2}.
Step 5 of 7: Substitute and isolate the odd parameter
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1−εm=2x−2(m2−8)1-\varepsilon m=2^{x-2}(m^2-8)
Detailed analysis

Substituting y=2x−1m+εy=2^{x-1}m+\varepsilon into the factored equation and dividing by 2x2^x gives 1+2x+1=2x−2m2+εm1+2^{x+1}=2^{x-2}m^2+\varepsilon m, equivalently 1−εm=2x−2(m2−8)1-\varepsilon m=2^{x-2}(m^2-8).