Is This a New Property I Have Found Pertaining to Mersenne Primes?

prime-numbers number-theory conjectures primality-test mersenne-numbers

As you observe, this is about representing $M_p$ by the quadratic form $a^2-ab+b^2$. That is the norm of the quadratic integer $a+b\omega$ where $\omega =\frac12(-1+i\sqrt3)$. A prime $q$ with $q\equiv1\pmod 3$ always has twelve representations by this form: there are two ideals of norm $q$ in $\Bbb Z[\omega]$ and each has six different generators. Indeed there is a formula for the number of representations $q$ by this in terms of the factorisation of $q$. The number of representations is only $12$ if $q=q'm^2$ where $q'$ is a prime congruent to $1$ modulo $3$ and the prime factors of $m$ are all congruent to $2$ modulo $3$. I can't see why it's impossible for a Mersenne number to have such a factorisation with $m>1$, but it does seem rather unlikely.

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