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 2 of 6: Solve the linear case
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n=1⟹x+(2+x)+(2−x)=0⟹x=−4n=1\Longrightarrow x+(2+x)+(2-x)=0\Longrightarrow x=-4
Detailed analysis

For n=1n=1, the equation is x+(2+x)+(2−x)=0x+(2+x)+(2-x)=0, which simplifies to x+4=0x+4=0. Thus x=−4x=-4 is an integer solution, so n=1n=1 works.