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 2020-05-19, 18:11 #12 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 26·29 Posts DHS-170-A For any given factorial n! greater than 4!, let v be the valuation of prime factor q of n!, where q >=5 Then we know that one of the (n! +/- q^v)/(q^v) will have a valuations of q that is 0 and the other will have a valuations of q that is greater than or equal to 0. Other than for q (for perhaps only one of the pair) the expressions will have no other common prime factors with n!. To-be-continued ...
 2020-05-19, 18:29 #13 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 185610 Posts DHS-180-A For any given factorial n! if (n! +/- k)/(k) are both positive integers which are both coprime to n! & (n! +/- k)/(k) < n^2, then (n! +/- k)/(k) are twin primes To-be-continued ... Last fiddled with by a1call on 2020-05-19 at 19:23
 2020-05-20, 02:59 #14 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 111010000002 Posts DHS-190-A There are infinitely many distinct twin-coprimes-to-n! (Please see post #9) of the form (n! +/- k)/(k), since for all integers n > 3 q = (n! - 2)/(2) & p = (n! + 2)/(2) Are distinct integers which are all coprime to n! and p = q + 2 Please see DHS-100-A To-be-continued ...
 2020-05-20, 03:56 #15 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 111010000002 Posts DHS-200-A Any given pair of twin primes q & p where p = q+2 can be expressed as (n! +/- k)/(k) for infinite integers n such that: n >= r where r is the greatest prime factor of (q+1) & k = n!/(q+1) It follows that there are infinite (not necessarily distinct) twin primes of the form (n! +/- k)/(k) To-be-continued ... Last fiddled with by a1call on 2020-05-20 at 04:25
2020-05-20, 05:53   #16
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

26×29 Posts

Quote:
 Originally Posted by a1call DHS-200-A Any given pair of twin primes q & p where p = q+2 can be expressed as (n! +/- k)/(k) for infinite integers n such that: n >= r where r is the greatest prime factor of (q+1) & k = n!/(q+1) It follows that there are infinite (not necessarily distinct) twin primes of the form (n! +/- k)/(k) To-be-continued ...
This is wrong. At least if the k has to be an integer.
Will see of I can come up with a correct version.
To be continued ...

Last fiddled with by a1call on 2020-05-20 at 05:54

2020-05-20, 06:21   #17
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

26×29 Posts

Quote:
 Originally Posted by a1call DHS-200-A Any given pair of twin primes q & p where p = q+2 can be expressed as (n! +/- k)/(k) for infinite integers n such that: n >= r where r is the greatest prime factor of (q+1) & k = n!/(q+1) It follows that there are infinite (not necessarily distinct) twin primes of the form (n! +/- k)/(k) To-be-continued ...
This version should work though not using/being a minimal n:

DHS-200-B Any given pair of twin primes q & p where p = q+2 can be expressed as (n! +/- k)/(k) for infinite integers n such that:
n >= r where r = (q+1)
&
k = n!/r

It follows that there are infinite (not necessarily distinct) twin primes of the form (n! +/- k)/(k)

To-be-continued ...[/QUOTE]

Last fiddled with by a1call on 2020-05-20 at 06:22

 2020-05-30, 22:41 #18 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 26·29 Posts A 79820 dd Prime example found using PFGW: (20577!-2652)/2652 Currently waiting for FactorDB to process the number. Stay tuned if not done. Last fiddled with by a1call on 2020-05-30 at 22:44
2020-05-31, 04:16   #19
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

185610 Posts

Quote:
 Originally Posted by a1call A 79820 dd Prime example found using PFGW: (20577!-2652)/2652 Currently waiting for FactorDB to process the number. Stay tuned if not done.
FTR the number was proven prime via PFGW by using the -tp flag, FactorDB says "Too big to be tested at the moment." for N+1 proof. I am not sure how temporary is the "at the moment" if at all.

2020-05-31, 04:23   #20
paulunderwood

Sep 2002
Database er0rr

23·3·139 Posts

Quote:
 Originally Posted by a1call FTR the number was proven prime via PFGW by using the -tp flag, FactorDB says "Too big to be tested at the moment." for N+1 proof. I am not sure how temporary is the "at the moment" if at all.
Congrats on the proof of your number!

 2020-05-31, 04:30 #21 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 35008 Posts Thank you Paul. I know it is small potatoes, but the size would be a top 10 (20 counting twins as double) if it were a twin prime which is what I am aiming for. I have a virtual-box instance which I run on 4 cores when I can, so in a few months who knows. Keeps me dreaming.

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