[kriesel]

Well, all positive integers >1 that is? 2¹-1 is 1, 2⁰-1 is 0, and 2ⁿ-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 f₁ = 2^a - 1 and f₂ = 2^b - 1 (and a cofactor). For example, 2⁶ - 1 = 2^(2*3) - 1 is divisible by (2²-1) and by (2³-1); 63 = 3 * 7 * cofactor 3, and 2¹⁵-1 = 2^(3*5) - 1 = 32767 = 7 * 31 * 151 = (2³ - 1) * (2⁵ - 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 x²-1=(x-1)(x+1)
or for a difference of cubes 2^3a-1, substitute 2^a=x into x³-1=(x-1)(x²+x+1)
2⁶ - 1 = (2³)² - 1 = (2³-1)(2³+1), because this is an x² - 1
Or
2⁶ - 1 = (2²)³ - 1 = (2²-1)(2⁴+2²+1), because this is an x³ - 1
One factor is sufficient to eliminate an exponent from consideration, but one can continue,
(2³+1) = (2+1)(2²-2¹+1) = 3 *3

Visual:
For M(n)=2ⁿ-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 2³⁰-1 = 1073741823. 30 = 2 * 3 * 5.
2²-1 = 3; 2³-1=7; 2⁵-1=31.
2³⁰-1 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 = 3² * 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