Elemental Patterns from the Erd{o}s Straus Conjecture

This paper makes the following conjecture: For every prime p there exists a positive integer x with p/4 ≤ x ≤ p/2 and a positive divisor d|x² so that either: (1) d ( 4x - p ) ≡ -px; or (2) d ≤ x and d ( 4x - p ) ≡ -x. Furthermore this paper proves that the solutions to these modular equations are in one-to-one correspondence with the solutions of the diophantine equation used in the Erd{o}s Straus conjecture.