Consider $2^n-1$. Based on checking a few small numbers for $n$ (in fact, the first ten natural numbers) we might "conclude" that "$n$ is prime if and only if $2^n-1$ is prime." However, further investigation shows that for $n=11$, $2^n-1$ is not prime. Thus our claim that "if $n$ is prime then $2^n-1$ is prime" is a near miss. However, we can prove its reverse, that "if $2^n-1$ is prime then $n$ is prime".

I am looking for some elementary number theoretic examples that both directions of our eventually fake biconditional are near misses.



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