Other ways to determine the validity of a public key Public key 3, 4 fits the equation of y^2 = x^2 + 7

Yet this comes up as an "invalid public key".   How is that invalid?  Sure, it deviates extremely from other numbers, but what other equations are being applied to the public key to determine its validity?
You got the formula wrong.

The correct formula is 

y2 = x3 + 7 mod N

where N is a very large prime.