Some other conjectures

[sweety439]

Originally Posted by Fan Ming

MM127+2 (aka 2^(2^127-1)+1) has a non-trivial factor: 886407410000361345663448535540258622490179142922169401.
It seems either New Mersenne (Wagstaff) conjecture will be false(if MM127 is prime) or M127 will be the last prime in the Catalan-Mersenne sequence(if MM127 is not prime).

I conjectured that 127 (not M127) is the largest number satisfying all three conditions in New Mersenne (Wagstaff) conjecture.

Also, how about test whether the numbers MW127 and WW127 is prime? (Wn = (2^n+1)/3) and other numbers like MW43, WW43, MM61, WM61, MW61, WW61? (MM43 and WM43 cannot be prime, since M43 is not prime, note that 127 is M7 and 43 is W7, and both M127 and W43 are primes)

[kriesel]

Originally Posted by sweety439

how about test whether the numbers MW127 and WW127 is prime? (Wn = (2^n+1)/3) and other numbers like MW43, WW43, MM61, WM61, MW61, WW61? (MM43 and WM43 cannot be prime, since M43 is not prime, note that 127 is M7 and 43 is W7, and both M127 and W43 are primes)

Probably very clear to you, but maybe not to some others:
Primality tests can not be performed on MM or MW or WW x where x > 32.
The software does not exist, to my knowledge. (Mlucas ~MM32; gpuowl ~MM31; mprime ~MM30 for AVX512)
The run times would be several years for gigadigit Mersennes or ~MM32, even on currently fastest available gpus.
Required ram and permanent storage also impose limits, on both primality testing and P-1 factoring, on gpus and cpus, estimated as less than MM37 and MM42 for current hardware feasible ram, and MM67 for existing file systems' capacity limits. https://www.mersenneforum.org/showpo...1&postcount=10
So I assume you mean how about heavily trial factor these.
How many gpu-months or -years are these worth?

[sweety439]

Originally Posted by Fan Ming

I agree with that. Most likely MM127 is not prime, thus primes in Catalan-Mersenne sequence end at M127.

I think it is not prime because it is corresponding to the number 5

Like the case of Fermat numbers:

F0=3 is prime, F1=5 is prime, F2=17 is prime, F3=257 is prime, F4=65537 is prime, but F5=4294967297=641*6700417 is composite, and Fn is composite at least for 5<=n<=32

In the case of Catalan-Mersenne numbers:

C0=2 is prime, C1=3 is prime, C2=7 is prime, C3=127 is prime, C4=170141183460469231731687303715884105727 is prime, but I think that C5 is composite, of course if C5 is composite, then Cn is composite for all n>=5

Fn = (F(n-1)-1)^2+1, F0 = 3
Cn = 2^C(n-1)-1, C0 = 2

There is another sequence highly related to these numbers, Double Mersenne numbers:

Define Dn = MM(n-th prime)

D1=7 is prime, D2=127 is prime, D3=2147483647 is prime, D4=170141183460469231731687303715884105727 is prime, but Dn is composite at least for 5<=n<=17

I think that Fn, Cn, and Dn are primes only for n<=4, and composite for n>4

(think about that: polynomial equations with degree n have algebraic solution if and only if n<=4)

[sweety439]

Originally Posted by JeppeSN

Nice factor!

The New Mersenne conjecture is rather silly, but it holds for small numbers, and it is maybe unlikely that large numbers will satisfy just two of the three criteria. It would be fun if MM127 were a counterexample, of course, but nobody thinks so.

/JeppeSN

I conjectured that all numbers > 127 satisfy at most one of the three criteria.