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