(04/04/2023, 11:00 PM)tommy1729 Wrote: (04/03/2023, 11:24 PM)tommy1729 Wrote: https://math.stackexchange.com/questions...ne-numbers
For the extended mersenne numbers that are so cruxial here, I am looking for special cases :
4 conditions for X :
**
X = an extended mersenne number
**
X = a product of two extended mersenne numbers A and B, where
A is a product of 2 primes C,D where C and D are extended mersenne numbers
and B is a prime.
Or equivalent X = BCD where B,C,D are primes and extended mersenne numbers.
**
X is squarefree.
**
(X + 1)/4 is a prime P and P is NOT an extended mersenne number.
OR simplified :
3 conditions for X :
X = BCD where B,C,D are primes and extended mersenne numbers.
**
X is squarefree.
**
(X + 1)/4 is a prime P and P is NOT an extended mersenne number.
( notice primes of the form 4n + 1 are not extended mersenne numbers )
regards
tommy1729
Example :
3*7*31 = 651
651+1=652
652/4=163 (prime)
SO dimension 652 is a serious candidate for the desired algebra.
Now working in dimension 652 by hand and/or pocket calculator seems alot of work.
The idea of computers occurs ...
7*19*31 = 4123
4123+1 = 4124
4124/4 = 1031 (prime)
1031 is another candidate.
At the moment I see no easy way to test these dimensions.
In particular without a computer.
Although I have some ideas with iterations.
regards
tommy1729
hmm 651 might be a problem.
651+1 = 652
652/4 = 163 ( prime )
but 163 = 2*81 + 1.
and 81 is an extended mersenne number.
lets look at the other one
7*19*31 = 4123
4123+1 = 4124
4124/4 = 1031 (prime)
1031 = 2*515 + 1
515 factors as 5 * 103 but 5 is not in the list so
515 = 2* 257 + 1
257 is prime
257 = 2 * 128 + 1
128 is a power of 2 so not in the list.
This means 4123 has passed the (first) test.
***
4123 is not a multiple of 652 so that is good.
Notice that multiples of 4123 might be suspect as well.
So maybe sieve multiples of 4123 since they might be extensions of the 4123 case.
And if 652 works , sieve 652 too maybe.
If we add that as an extra rule, then the density of candidates goes down rapidly and complicated , at least probably.
Since it is complicated it is hard to say but it seems it then goes down as a fraction of the density of prime twins.
But I need to think more about it.
I did assume the expected density of prime twins here, although the actual might also be true. I do not insist on going in that kind of number theory since
1) it is not exactly about prime twins , just similar heuristics
2) statistical number theory can be weird.
Some patterns suggest counterintuive results , like
https://math.stackexchange.com/questions...ain-primes
3) actually sophie germain primes are more related because of the 2x + 1 rule.
Conclusion
maybe maybe 652
maybe 4123
although we know alot about their structures, construction is hard.
knowledge is mainly about a part of the multiplication table.
But the actual algebra is alot harder.
Dimension 4123 is huge !
***
about this condition ...
(X + 1)/4 is a prime P and P is NOT an extended mersenne number.
( notice primes of the form 4n + 1 are not extended mersenne numbers )
This implies P of the form 4n + 1 could work since it is never an extended mersenne number.
therefore
(X+1)/4 = P = 4n + 1
X + 1 = 16n + 4
X = 16n + 3
are interesting candidates.
652 is not of that form.
4123 is not of that form.
just some ideas
***
regards
tommy1729
