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Old 2020-05-19, 18:11   #12
a1call
 
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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 ...
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Old 2020-05-19, 18:29   #13
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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
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Old 2020-05-20, 02:59   #14
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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 ...
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Old 2020-05-20, 03:56   #15
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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
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Old 2020-05-20, 05:53   #16
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Quote:
Originally Posted by a1call View Post
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
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Old 2020-05-20, 06:21   #17
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Quote:
Originally Posted by a1call View Post
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
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Old 2020-05-30, 22:41   #18
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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
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Old 2020-05-31, 04:16   #19
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Quote:
Originally Posted by a1call View Post
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.
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Old 2020-05-31, 04:23   #20
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Quote:
Originally Posted by a1call View Post
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!
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Old 2020-05-31, 04:30   #21
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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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