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