Problem 4

Determine all pairs

(x,y)

of integers such that

1+2^{x}+2^{2x+1}=y^{2}

Step 4 of 7: Separate the odd part

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x\ge3,\qquad y=2^{x-1}m+\varepsilon,\quad m\text{ odd},\quad \varepsilon=\pm1

Detailed analysis

Since $1+2^{x+1}$ is odd, the total 2-adic valuation on the right is $x$ . The two even factors differ by $2$ , so one has valuation exactly $1$ and the other carries the remaining power; this forces $x\ge3$ and lets us write $y=2^{x-1}m+\varepsilon$ with $m$ odd and $\varepsilon=\pm1$ (take $y>0$ , since changing the sign of $y$ preserves the equation).