How to check whether Mersenne number is a prime or composite by using theorem $q = 2kp + 1$?

Question

How to know that whether any Mersenne Number, $M_{m}$ whether is a prime, or composite?

I have learnt that $M_{m}=2^m-1$ , for $m$ is any positive interger.

And there is a theorem, says that :

if $p$ is odd prime, then any divisor of Mersenne number is of the form $2kp + 1$, where $k$ is a positive integer.

Then I have carried out an example by plugging $m=37$,

$M_{37}=2^{37}-1=137438953471$ (although I know that it is not a prime), and

we have $q=2kp+1=2k(37)+1 =74k+1$

For

$k=2,q=149\longrightarrow$ Since $149$ is not the divisor for $M_{37}$, so this implies that $M_{37}$ is currently a prime.

$k=3,q=223\longrightarrow$ Since $223$ is the divisor for $M_{37}$, so this implies that $M_{37}$ is immediately not a prime.

$\therefore M_{37}$ is not a prime.

Do my reasoning is correct? Can anyone give me a comment about it? Also, can I start with using $k=1$ ?

Will be very thankful !

Answer 1

Yes, it is correct, but you should no say that “$M_{37}$ is currently a prime”; at this moment, you simply do not know whether or not it is a prime number.

And, yes, you can start with $k=1$. Take $M_{11}(=2047)$, for instance. If $k=1$, then $q=23$. And, in fact, $23\mid2047$.