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

Question

While playing with Mersenne numbers, I found the following property distinguishing Mersenne prime numbers from Mersenne composite numbers.

A Mersenne number, $\text{M}p$, is a number of the form $2^p - 1$, where $p$ is prime.

Property

For $p > 2$, Mersenne primes can be expressed as \begin{align*} \text{M}p = \frac{a^3 + b^3}{a + b}\text{,} \end{align*} where $a$ and $b$ are integers, $a \neq -b$, with exactly $12$ different solutions. So far, also $\operatorname{gcd}(a,b)=1$ holds for Mersenne primes.

Mersenne composites have either no integer solution or more than $12$ solutions ($24$ so far). Also $\operatorname{gcd}(a,b)=1$ does not hold if the integer solutions exist so far.

Examples

\begin{align*} \text{M}5 &= \frac{6^3 + 5^3}{11} = 31 \\ \text{M}7 &= 7^3 - 6^3 = 127 \end{align*}

The M11 has no integer solution for $(a,b)$.

The M37 has 24 solutions and also $\operatorname{gcd}(a,b)=1$ does not hold.

Remarks

• Except the M2, twelve solutions exist for each Mersenne prime. If $(a,b)$ is a solution, then also $(-a,-b)$, $(b,a)$, and $(-b,-a)$ are.

• Since \begin{align*} \frac{a^3 + b^3}{a + b} = a^2 -ab + b^2 \text{,} \end{align*} each Mersenne prime has an ellipse intersecting integer grid associated with it. For example, $-a^2 + ab - b^2 + 127 = 0$ is the ellipse for M7.

Results

Solutions for the first few Mersenne numbers: $$\begin{matrix} p & \text{M}p & (a,b) \\ \hline 2 & 3 & (1,2) \\ 3 & 7 & (1,-2), (1,3), (2,3) \\ 5 & 31 & (1,-5), (1,6), (5,6) \\ 7 & 127 & (6,-7), (6,13), (7,13) \\ 11 & 2047 & \text{no solution} \\ 13 & 8191 & (1,-90), (1,91), (90,91) \\ 17 & 131071 & (6,-359), (6,365), (359,365) \\ 19 & 524287 & (83,-679), (83, 762), (679, 762) \\ 23 & 8388607 & \text{no solution} \\ 29 & 536870911 & \text{no solution} \\ 31 & 2147483647 & (4698, 43813), (4698,48511), (43813, 48511) \\ 37 & 137438953471 & \text{24 solutions} \\ 41 & 2199023255551 & \text{no solution} \\ 43 & 8796093022207 & \text{no solution} \\ ... & ... & ... \\ \end{matrix}$$

I have verified the conjecture using WolframAlpha for all $p$ below 100.

Question

Can you confirm this result? Is this known? Any feedback is welcome.

Answer 1

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.