Elementary Patterns from the Erdos-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 | x
2
so that either
d ≡ −px mod (4x − p) or d ≤ x and d ≡ −x mod (4x − p). 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˝os-Straus
conjecture.