Many properties of primes are very difficult to determine and prove. Here's an exploration that's within reach of many of us:
What is interesting about the difference of the squares of most any two primes? In other words, what is interesting about p12-p22 for most primes p1 and p2? When does this property hold? Prove your assertion.
I got the idea for this puzzle/exploration from Standup Maths and adapted it to make it harder!