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Old 2020-07-07, 04:41   #3
Nov 2016

281910 Posts

Originally Posted by Fan Ming View Post
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)

Last fiddled with by sweety439 on 2020-07-07 at 04:52
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