Problem 4

Determine all positive integers

n

for which

x^n+(2+x)^n+(2-x)^n=0

has an integer solution.

Step 6 of 6: Derive the 2-adic contradiction

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v_2(y^n)\ge n>1\quad\text{but}\quad v_2(-2T)=1\Longrightarrow\text{contradiction}

Detailed analysis

Because $T$ is odd, the right side $-2T$ has exactly one factor of $2$ . But $y$ is even, so $y^n$ has at least $n>1$ factors of $2$ . This contradiction rules out every odd $n>1$ . Together with the earlier cases, the only answer is $n=1$ .