Let $k$ be a positive integer. Prove that: (a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite. (b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.
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Tags: number theory
Let $k$ be a positive integer. Prove that: (a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite. (b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.