Mersenne Number whose Index is Mersenne Prime

Conjecture

A Mersenne number whose index is a Mersenne prime is itself a Mersenne prime.

Refutation

The smallest counterexample is $M_{8191}$.

We have that $M_{13} = 2^{13} - 1 = 8191$ is a Mersenne prime.

But $M_{8191} = 2^{8191} - 1$ is composite.

$\blacksquare$

Historical Note

According to David Wells, in his Curious and Interesting Numbers, 2nd ed. of $1997$, this conjecture was made by Eugène Charles Catalan.

The composite nature of $2^{8191} - 1$ was reported on by Raphael Mitchel Robinson in $1954$, in an article in the Proceedings of the American Mathematical Society:

The corresponding Mersenne number was actually tested in $1953$ by D.J. Wheeler on the Illiac, at the University of Illinois, one hundred hours of machine time being required.

Sources

• Oct. 1954: Raphael M. Robinson: Mersenne and Fermat Numbers (Proc. Amer. Math. Soc. Vol. 5, no. 5: pp. 842 – 846)  www.jstor.org/stable/2031878

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $8191$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{8191} - 1$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8191$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{8191} - 1$