"WHY ARE 1/2 OF THE NON ZERO NUMBERS IN THE (MOD P) SYSTEM PERFECT SQUARES AND THE OTHER HALF NOT?"
This is what led me to realize the part about how half of the non zero numbers in (mod p) system, (which is nine since not counting zero there are 18 integers):
since I was able to prove through using a mod 19 multiplication table that every non zero number a, in the set of mod 19, (1,2,3,.....,18) has a multiplicative inverse, ( proven by 2 ways: 1, that the GCD of both a (a in the set of mod 19, (1, 2, 3,...........,18) and m (the module) is prime equaling a GCD of 1, and that every row and column has a 1.
So since a modular multiplicative inverse of a mod p exists, the operation of division by a mod p can be defined as multiplying by the inverse, ( which is the same thing as dividing in the field of mod p.
So then I was looking for more relationships between the integers a in mod p systems (using the mod 19 multi. table which I'll include below) and noticed if I circled all the ones that there was a Main and Opposite lines of diagonal symmetry (the two diagonal black lines on the table). So on each side of the line the 1's were all reflective of each other ( the exact same on each side). which then made me think of perfect squares (multiplying the same number by itself which kinda made me think of symmetry) so I looked along the main diagonal line (which touched 18 numbers) the first 9 were perfect squares (the rest of the 9 numbers were repeats of the first 9) so the 9 perfect squares in any prime mod are: [(1, 4, 9, 16, 6, 17, 11, 7, 5) {Mod 19}].
6(mod 19) =25; 17(mod 19)=36; 11(mod 19)=49; 7(mod 19)=64; and
5(mod 19) =81
Thanks if anyone can help:)
