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

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 !

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, $$23\mid2047$$.
• in fact $k\equiv 0,-p\pmod {3,4}$ May 5 at 17:33