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.
Best Answer
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.