Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4

Abstract

The Erdos-Straus conjecture (ESC) concerns the representation of the fraction 4/P, where P is a prime number, as a sum of three positive unit fractions. The focus here is on the case when P is congruent to 1 modulo 4. Two constructive approaches are proposed. Method ED1 is based on a factorization identity and leads to a nonlinear parameterization in P, which requires divisor enumeration and local filtering. Method ED2 yields a linear system in P for the parameters (delta, b, c), describing the solution set as an affine lattice of finite index in Z^3. The central result states that for every prime P congruent to 1 modulo 4 there exists a representation: 4/P = 1/A + 1/(bP) + 1/(cP), where the triple (delta, b, c) in N^3 is constructed explicitly by method ED2. In addition, algorithms for transforming solutions (convolution and anti-convolution) are introduced, and large-scale computational verification confirms the correctness and efficiency of the proposed methods.