Well, all positive integers >1 that is? 2

^{1}-1 is 1, 2

^{0}-1 is 0, and 2

^{n}-1 for n<0 is a negative real, not an integer, so can't be a Mersenne prime.

For any Mersenne number formed from a composite exponent, the resulting Mersenne number is composite.

https://primes.utm.edu/notes/proofs/Theorem2.html Some of the Mersenne number's factors are easily found by deriving them from the prime factors of the composite exponent. The Mersenne number of a composite exponent a*b is not only composite, it is a repdigit, with M(a*b) a repdigit in base 2

^{a} with b digits 2

^{a}-1, and in base 2

^{b} with a digits 2

^{b}-1.

Here are 3 ways of looking at it.

**Numerical**:

If n = a * b, 1<a<n and a<b<n, integers, the Mersenne number has factors f

_{1} = 2

^{a} - 1 and f

_{2} = 2

^{b} - 1 (and a cofactor). For example, 2

^{6} - 1 = 2

^{(2*3)} - 1 is divisible by (2

^{2}-1) and by (2

^{3}-1); 63 = 3 * 7 * cofactor 3, and 2

^{15}-1 = 2

^{(3*5)} - 1 = 32767 = 7 * 31 * 151 = (2

^{3} - 1) * (2

^{5} - 1) * cofactor 151.

Algebraic:

Note in the proof section of

https://primes.utm.edu/notes/proofs/Theorem2.html and that r and s can be swapped. For a difference of squares such as 2

^{2a}-1, substitute 2

^{a}=x into x

^{2}-1=(x-1)(x+1)

or for a difference of cubes 2

^{3a}-1, substitute 2

^{a}=x into x

^{3}-1=(x-1)(x

^{2}+x+1)

2

^{6} - 1 = (2

^{3})

^{2} - 1 = (2

^{3}-1)(2

^{3}+1), because this is an x

^{2} - 1

Or

2

^{6} - 1 = (2

^{2})

^{3} - 1 = (2

^{2}-1)(2

^{4}+2

^{2}+1), because this is an x

^{3} - 1

One factor is sufficient to eliminate an exponent from consideration, but one can continue,

(2

^{3}+1) = (2+1)(2

^{2}-2

^{1}+1) = 3 *3

**Visual**:

For M(n)=2

^{n}-1 where n=a b, M(n) has factors 2

^{a}-1 and 2

^{b}-1. Such a number is a repdigit in base 2

^{a} and in base 2

^{b}.

It's similar to being able to tell at a glance that numbers like 99, 999, 9999, 99999, 999999 are divisible by nine (and by 11, 111, 1111 = 11 x 101, 11111, and 111111 = 11 x 10101 = 111 x 1001 respectively).

Consider the small example 2

^{30}-1 = 1073741823. 30 = 2 * 3 * 5.

2

^{2}-1 =

**3**; 2

^{3}-1=

**7**; 2

^{5}-1=

**31**.

2

^{30}-1 in binary is 30 ones bits consecutively, which can be grouped into equal-sized groups in several ways;

first the digits that form prime factors (

**3, 7, 31**);

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 = 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 base 4;

111 111 111 111 111 111 111 111 111 111 = 7 7 7 7 7 7 7 7 7 7 base 8;

11111 11111 11111 11111 11111 11111 = 1F 1F 1F 1F 1F 1F base 32 (sort of; using multidigit hexadecimal to represent the base-32 individual digits here and for large bases below, rather than pedantically adopt different large ASCII or UNICODE symbol sets for one or two lines of uses);

these digits are composite factors (2

^{2*3}-1 =63, 2

^{2*5}-1= 1023, 2

^{3*5}-1 = 32767);

111111 111111 111111 111111 111111 = 3F 3F 3F 3F 3F base 64;

1111111111 1111111111 1111111111 = 3FF 3FF 3FF base 1024;

111111111111111 111111111111111 =7FFF 7FFF base 32768.

Fully factoring it such as via

https://www.alpertron.com.ar/ECM.HTM yields

1073 741823 =

**3**^{2} *

**7** * 11 *

**31** * 151 * 331

It has Mersenne primes among its prime factors.

(extended with input from Batalov, specifically the polynomial factoring section)

Top of reference tree:

https://www.mersenneforum.org/showpo...22&postcount=1