mersenneforum.org Mersenne Primes p which are in a set of twin primes is finite?
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 2017-08-08, 17:05 #1 carpetpool     "Sam" Nov 2016 5178 Posts Mersenne Primes p which are in a set of twin primes is finite? I want to bring up the of the Mersenne Prime 2^n-1 being the second prime p+2 in a twin prime pair {p, p+2} are there finitely many Mersenne Primes which hold this condition (this is the same as primes p such that 2^p-1 and 2^p-3 are prime). First off 2^n-1 and 2^n+1 cannot both be prime for n > 2, therefore we only focus on 2^n-1 and 2^n-3 both being primes. Second, if 2^n-1 and 2^n-3 are both prime, n must be prime because if n is composite = ab, then 2^n-1 = (2^a-1)*(1 + 2^a + 2^(2*a) + 2^(3*a) .... + 2^(b*a-a) Third, if 2^n-1 and 2^n-3 are both prime, n = 1 (mod 4), because if n = 3 (mod 4), 2^n-3 = 0 (mod 5) cannot be a prime. This follows from 2^(4*n+3) = 3 (mod 5) - 3 = 0 (mod 5). The only known exponents for which 2^n-1 and 2^n-3 are 3 and 5 (up to the Same Limit the Mersenne Numbers were tested). This is conjectured to be finite unless anyone brings up an arguments as to maybe why not. Are there any more restrictions to this? Thanks for help.
2017-08-08, 20:31   #2
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

2×3×23×61 Posts

Quote:
 Originally Posted by carpetpool I want to bring up the of the Mersenne Prime 2^n-1 being the second prime p+2 in a twin prime pair {p, p+2} are there finitely many Mersenne Primes which hold this condition (this is the same as primes p such that 2^p-1 and 2^p-3 are prime). First off 2^n-1 and 2^n+1 cannot both be prime for n > 2, therefore we only focus on 2^n-1 and 2^n-3 both being primes. Second, if 2^n-1 and 2^n-3 are both prime, n must be prime because if n is composite = ab, then 2^n-1 = (2^a-1)*(1 + 2^a + 2^(2*a) + 2^(3*a) .... + 2^(b*a-a) Third, if 2^n-1 and 2^n-3 are both prime, n = 1 (mod 4), because if n = 3 (mod 4), 2^n-3 = 0 (mod 5) cannot be a prime. This follows from 2^(4*n+3) = 3 (mod 5) - 3 = 0 (mod 5). The only known exponents for which 2^n-1 and 2^n-3 are 3 and 5 (up to the Same Limit the Mersenne Numbers were tested). This is conjectured to be finite unless anyone brings up an arguments as to maybe why not. Are there any more restrictions to this? Thanks for help.
the restriction for exponents of form 1 mod 4 also comes from mersenne primes greater than 7, are 7 and 31 mod 120, the restriction that 2^n-3 also be prime restricts to exponents that give rise to 31 mod 120.

2017-08-10, 13:36   #3
Dr Sardonicus

Feb 2017
Nowhere

6,221 Posts

Quote:
 Originally Posted by carpetpool I want to bring up the of the Mersenne Prime 2^n-1 being the second prime p+2 in a twin prime pair {p, p+2} are there finitely many Mersenne Primes which hold this condition (this is the same as primes p such that 2^p-1 and 2^p-3 are prime).
The question of 2^p - 3 and 2^p - 1 both being prime, doesn't seem very interesting. After all, 2^p - 1 is very seldom prime.

The question of when 2^n - 3 alone might be prime may be of some interest in its own right. It wouldn't surprise me at all if someone had compiled a factor table for n into the hundreds, and a list of pseudoprimes for larger n's.

By way of keeping in practice with this sort of thing, I note the following:

Values of n < 1000 for which 2^n - 3 tests as a pseudoprime:

n = 3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850

Prime values of 2^n - 3 seem to occur for composite n much more often than for prime n.

2^p - 3 is (pseudo)prime, but 2^p - 1 is not, for the primes p = 29 and 233.

Recurrent prime divisors p < 100 of 2^n - 3. Primes followed by an asterisk only divide 2^n - 3 for composite exponents n.

5 (n = 4*k + 3)
11* (n = 10*k + 8)
13* (n = 12*k + 4)
19 (n = 18*k + 13)
23 (n = 11*k + 8)
29 (n = 28*k + 5)
37* (n = 36*k + 26)
47 (n = 23*k + 19)
53 (n = 52*k + 17)
59* (n = 58*k + 50)
61* (n = 60*k + 6)
67* (n = 66*k + 39)
71 (n = 35*k + 16)
83* (n = 82*k + 72)
97 (n = 48*k + 19)

Non-divisors p of 2^n - 3: There are the obvious cases p = 2 and 3. For p > 3, we have the following: If 2 is a q-th power residue of the prime p but 3 is not, then no power of 2 can be congruent to 3 (mod p). In the case q = 2 we can say this is the case for p congruent to 7 or 17 (mod 24), e.g. p = 7, 17, 31, 41, 79, 89. An example with q = 3 is p = 43.

2017-08-10, 13:47   #4
paulunderwood

Sep 2002
Database er0rr

23×3×11×17 Posts

Quote:
 Originally Posted by Dr Sardonicus The question of when 2^n - 3 alone might be prime may be of some interest in its own right. It wouldn't surprise me at all if someone had compiled a factor table for n into the hundreds, and a list of pseudoprimes for larger n's.

I was interested in 2-PRP being enough for these PRPs. In general 2^n-2^k-1. Most general a-PRP for:

"For integers a>1, s>=0, all r>0, all t>0, odd and irreducible {a^s\times\prod{(a^r-1)^t}}-1 is a-PRP, except for the cases a^2-a-1 and a-2 and a-1 and -1."

This includes a^2-2, for which I have done a cursory check for a < 10^13.

Last fiddled with by paulunderwood on 2017-08-10 at 13:53

2022-07-14, 02:29   #5
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1110011000102 Posts

Quote:
 Originally Posted by carpetpool I want to bring up the of the Mersenne Prime 2^n-1 being the second prime p+2 in a twin prime pair {p, p+2} are there finitely many Mersenne Primes which hold this condition (this is the same as primes p such that 2^p-1 and 2^p-3 are prime). First off 2^n-1 and 2^n+1 cannot both be prime for n > 2, therefore we only focus on 2^n-1 and 2^n-3 both being primes. Second, if 2^n-1 and 2^n-3 are both prime, n must be prime because if n is composite = ab, then 2^n-1 = (2^a-1)*(1 + 2^a + 2^(2*a) + 2^(3*a) .... + 2^(b*a-a) Third, if 2^n-1 and 2^n-3 are both prime, n = 1 (mod 4), because if n = 3 (mod 4), 2^n-3 = 0 (mod 5) cannot be a prime. This follows from 2^(4*n+3) = 3 (mod 5) - 3 = 0 (mod 5). The only known exponents for which 2^n-1 and 2^n-3 are 3 and 5 (up to the Same Limit the Mersenne Numbers were tested). This is conjectured to be finite unless anyone brings up an arguments as to maybe why not. Are there any more restrictions to this? Thanks for help.
For any given k, the number of twin primes (2^n+k-1,2^n+k+1) should be finite

If k is odd, then 2^n+k-1 and 2^n+k+1 are even and cannot be prime, if k is divisible by 3, then one of 2^n+k-1 and 2^n+k+1 is divisible by 3 (since 2^n cannot be divisible by 3) and cannot be prime

Also a special case for twin primes: 2^n-9 and 2^n-11 cannot be both primes if n > 4, since if 2^n-9 is prime and n > 4, then n is odd (since for even n, 2^n-9 = (2^(n/2)-3) * (2^(n/2)+3) is not prime if n > 4), but if 2^n-11 is prime, then n is even (since for odd n, 2^n-11 is divisible by 3 and cannot be prime)

Conjectures:

* 2^n-1 and 2^n-3 cannot be both primes if n > 5
* 2^n+1 and 2^n+3 cannot be both primes if n > 16
* 2^n-3 and 2^n-5 cannot be both primes if n > 150
* 2^n+3 and 2^n+5 cannot be both primes if n > 3
* There is no n such that 2^n-7 and 2^n-9 are both primes
* 2^n+7 and 2^n+9 cannot be both primes if n > 30
* 2^n+9 and 2^n+11 cannot be both primes if n > 23

etc.

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