MathLabs

Problem 4

Determine all positive integers nn for which xn+(2+x)n+(2−x)n=0x^n+(2+x)^n+(2-x)^n=0 has an integer solution.
Step 1 of 6: Exclude even exponents
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n even⟹xn+(2+x)n+(2−x)n>0n\text{ even}\Longrightarrow x^n+(2+x)^n+(2-x)^n>0
Detailed analysis

If nn is even, each of xnx^n, (2+x)n(2+x)^n, and (2−x)n(2-x)^n is nonnegative. They cannot all be zero simultaneously, so their sum is positive. Therefore any possible nn must be odd.