View Single Post
Old 2021-01-02, 00:48   #2
a1call
 
a1call's Avatar
 
"Rashid Naimi"
Oct 2015
Remote to Here/There

111110101102 Posts
Default

Hint #2:
For all positive integers p where valuation(p-1,2)==1
,
Mod( ((p-1)/2)! , p) == 1 || Mod( ((p-1)/2)! , p-1)

if and only if p is prime

Hint #3:
For all primes p where valuation(p-1,2)>1
,
there exist coprime integers a and b such that a+b=p && p | ab-1
(Not all but some composites satisfy this condition as well such as 25 where a=7 and b= 18)

ETA: being off by 100 years isn't that bad really, considering that numbers are unlimited and I could potentially be off by an indefinitely larger value.
ETA II
Hint #4:

For all primes p where valuation(p-1,2)>1
,
There are coprime integers a and b such that
Mod(( (p-1)/2)!, p) == a
and
Mod(( (p-2)! / (p-1)/2)!, p) == b
and a+b=p
and 1<a<b<p

Last fiddled with by a1call on 2021-01-02 at 01:29
a1call is offline   Reply With Quote