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

E. Dyachenko

TL;DR

This work tackles the Erdos–Straus conjecture for primes $P\equiv1\pmod4$ by introducing two constructive approaches, ED1 (a nonlinear in $P$ parameterization via a factorization identity) and ED2 (a linear-in-$P$ framework yielding an affine lattice in $(\delta,b,c)$). It proves a constructive existence theorem: for every such prime, there exists a representation $4/P=1/A+1/(bP)+1/C$ with $C=cP$ and $(\delta,b,c)$ explicit from ED2, and it develops convolution/anticonvolution to connect the two methods. The paper establishes affine-lattice structure, density estimates, and convergence results for ED2, and validates the theory with large-scale computational verification, including examples up to several thousands in $P$. Overall, the work provides a rigorous, computation-backed pathway to a constructive ES decomposition in the $P\equiv1\pmod4$ case and outlines a program toward a canonical, unified parameterization.

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.

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

TL;DR

This work tackles the Erdos–Straus conjecture for primes by introducing two constructive approaches, ED1 (a nonlinear in parameterization via a factorization identity) and ED2 (a linear-in- framework yielding an affine lattice in ). It proves a constructive existence theorem: for every such prime, there exists a representation with and explicit from ED2, and it develops convolution/anticonvolution to connect the two methods. The paper establishes affine-lattice structure, density estimates, and convergence results for ED2, and validates the theory with large-scale computational verification, including examples up to several thousands in . Overall, the work provides a rigorous, computation-backed pathway to a constructive ES decomposition in the case and outlines a program toward a canonical, unified parameterization.

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.

Paper Structure

This paper contains 105 sections, 72 theorems, 292 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Ordering
  4. Case P ≡ 3 (mod 4) — explicit decompositions
  5. Estimate for the minimal denominator
  6. Literature Review on Parameterization Approaches to ES
  7. Modular identities
  8. Factorization schemes
  9. Linear forms and lattices
  10. Analytic and computational methods
  11. Generalizations and educational sources
  12. Notation
  13. Parameterization of the Main Equation of the ES Conjecture in the Case B ≠ 0 (mod P)
  14. Algebraic transformation (reconstructed derivation).
  15. Complete parameterization via divisors of 1
  16. ...and 90 more sections

Key Result

lemma 3.1

Let $P>2$ and $4 \nmid P$, and Then the strict inequalities hold.

Figures (3)

  • Figure 1: Console output screenshot from search2521_new.py for parameter $P=2521$: the table displays computed values $(A, B, C)$, coefficients $c$, $u$, the product $uv = c'$, and validation checks. All rows pass successfully (indicated by the “OK” columns).
  • Figure 2: Branching scheme in anticonvolution ED2$\rightarrow$ED1 and invertibility conditions
  • Figure 3: Geometric assembly for parameter $P$: the diagram shows lattice point categories (even-even, odd-odd, primitive, solution), the triangular region $F(P)$, vertical strip $m$, target window, and intersections corresponding to valid solutions. This illustration supports the visual analysis of coverage and parametrization in Appendix D.

Theorems & Definitions (148)

  • lemma 3.1
  • proof
  • Theorem 6.1
  • proof
  • lemma 6.2: Sum and discriminant
  • proof : Идея доказательства
  • lemma 6.3: Back‑test for $(u,v)$ with fixed prime $P$
  • proof : Идея доказательства
  • lemma 6.4: Quadratic reparameterization
  • proof : Идея доказательства
  • ...and 138 more