Bounded Gateway Parameters for the Erdos-Straus Conjecture

Gateway Decompositions via Divisors of N^2

Overview

The Erdos-Straus conjecture (1948) asserts that for every integer n >= 2, the equation

4/n = 1/x + 1/y + 1/z

has a solution in positive integers x, y, z.

This repository contains the paper, verification code, and figures for a unified algebraic approach to the prime case. Every prime p = 3 (mod 4) is handled classically with A = 1. For every prime p = 1 (mod 4), a single auxiliary parameter A produces an explicit decomposition: three algebraic propositions cover all but a density-zero residual class, and a short search over prime values of A resolves the rest.

Verification to 10^11 (all 4,118,054,813 primes) succeeds with just 32 values of A for the p = 1 (mod 4) primes - the value A = 3 for the algebraic classes plus 31 distinct gateway values, all = 3 (mod 4) and all <= 359.

Key Result

Theorem. For every prime p = 1 (mod 4) in the residual class, there exists a prime A <= 359 with A = 3 (mod 4), 4 | (p + A), and a divisor d of ((p+A)/4)^2 satisfying d = -p^2 * 4^{-1} (mod A), such that

x = (p+A)/4,  y = (p*x + d)/A,  z = p*x*y/d

are positive integers giving 4/p = 1/x + 1/y + 1/z.

The critical insight is that the integrality condition requires d | N^2 (where N = (p+A)/4), which is strictly weaker than d | N. A nontrivial fraction of the hardest primes use a divisor d > N that is invisible under the stronger condition.

Bounded-A Phenomenon

The maximum A grows extremely slowly with scale, and does not grow at all across the most recent decade:

Limit	max A	Increase
10^6	79	-
10^7	167	+88
10^8	239	+72
10^9	239	+0
10^10	359	+120
10^11	359	+0

The record-setting prime is p = 3,807,728,761 (A = 359, d = 1935), which lies below 10^10 - nothing in (10^10, 10^11] beats it. Across 4.1 billion primes, only 47 require A >= 199 and only 5 require A >= 251. This slow, eventually-flat growth motivates the bounded-A conjecture: max A is absolutely bounded. If true, the full Erdos-Straus conjecture follows.

Paper

The main paper (erdos_straus_gateway.tex) contains:

• Section 3: The Gateway Decomposition theorem and the d | N^2 lemma
• Section 4: Algebraic existence proofs covering ~97.1% of all primes (at 10^6, rising with the bound)
• Section 5: Computational verification for the remaining residual class (Case B QR7 primes), including an Anatomy of the hardest prime - a discrete-log proof of why p = 3,807,728,761 forces A = 359 (its N^2 divisors fill every unit mod 359)
• Section 6: Discussion of the bounded-A phenomenon and the path to a full proof

The residual class - p = 1 (mod 24), a quadratic residue mod 7, with every prime factor of (p+3)/4 congruent to 1 (mod 3) - is related to but not identical with the classical Mordell subset {1, 121, 169, 289, 361, 529} (mod 840); Remark 4.5 distinguishes them with explicit examples.

Companion Document

• unconditional_bound.tex - An unconditional density bound E(X) = O(X/(log X)^{3/2}) on the count of primes not covered by any known algebraic or gateway decomposition, via the sharpness of gateway A = 7 and a half-dimensional Selberg sieve. Also proves A = 7 is the unique sharp gateway among the original 28-candidate list and discusses the character-sum barrier to an unconditional finiteness result.

Verification

Run the self-contained verification script (Python 3.6+, standard library only):

python verify.py              # all primes to 10^6  (~10 seconds)
python verify.py 10000000     # to 10^7             (~2 minutes)
python verify.py 100000000    # to 10^8             (~30 minutes)
python verify.py 1000000000   # to 10^9             (~90 minutes, ~1 GB RAM)

Reproduce the hardest-prime analysis (Section 5.4):

python verify_hardest_prime.py   # checks every claim for p = 3,807,728,761

Example output at 10^6:

Erdos-Straus Gateway Verification to 1,000,000
============================================================
Category                                            Count        %
-------------------------------------------------------------------
  p = 2                                                 1    0.001%
  p = 3 (mod 4)  [Prop. 4.1]                      39,322   50.093%
  p = 5 (mod 8)  [Prop. 4.2a]                     19,623   24.998%
  p = 17 (mod 24) [Prop. 4.2b]                     9,820   12.510%
  Case A  [Prop. 4.2c]                             5,192    6.614%
  Case B NQR7  [Prop. 4.3]                         2,271    2.893%
  Case B QR7  [Thm. 5.1]                           2,269    2.891%
-------------------------------------------------------------------
  PROVEN                                          78,498  100.0000%
  OPEN                                                 0    0.00000%

Max A needed: 79
ALL 78,498 PRIMES TO 1,000,000 VERIFIED.

For the full-scale segmented, checkpointed verification to 10^10 / 10^11 (16-worker pool, GPU sieve if CuPy is present):

python sessions/session20_corrected_10B.py 10000000000  results/session20_checkpoint.json   # 10^10
python sessions/session20_corrected_10B.py 100000000000 results/session21_checkpoint.json   # 10^11

Each run validates against three independent baselines (the full A-distribution at 10^6, the residual count at 10^7, and the residual count and max A at 10^9) before extending the range, and verifies every solution by direct evaluation of the identity. The results/ directory holds the resulting checkpoints with per-decade milestone snapshots.

Figures

Generate the paper figures (requires matplotlib):

python collect_stats.py     # regenerate stats_1M.json (10^6 sample)
python generate_figures.py  # fig1-fig5 as PDF and PNG
python generate_banner.py   # the README banner

Figure	Description
fig1_coverage	Prime classification hierarchy (pie charts)
fig2_A_distribution	Gateway parameter A distribution (bar + cumulative)
fig3_max_A	Max A growth across scales (10^3 to 10^11)
fig4_d_over_N	d/N ratio distribution showing the N^2 extension
fig5_gateway_diagram	Conceptual schematic of the decomposition

The Open Problem

Since every prime outside the residual class is covered algebraically, the conjecture reduces to:

For every prime p = 1 (mod 24) that is a quadratic residue mod 7 with all prime factors of (p+3)/4 congruent to 1 (mod 3), does there exist a bounded prime A = 3 (mod 4) such that ((p+A)/4)^2 has a divisor in the residue class -p^2 * 4^{-1} (mod A)?

This is a question about equidistribution of divisors in residue classes for structured integers, connected to work of Hooley and Tenenbaum.

Data Provenance

The verification pipeline was corrected in July 2026: an earlier run of the 10^10 stage inverted the Case A / Case B classification and searched the wrong set of primes. All figures in this repository derive from the corrected, baseline-validated pipeline (sessions/session20_corrected_10B.py) and are cross-checked against the checkpoints in results/. Section 5 of the paper documents the correction.

Citation

@article{erdos-straus-gateway-2026,
  title={Bounded Gateway Parameters for the {Erd\H{o}s--Straus} Conjecture},
  author={\'O Murch\'u, Macdara},
  year={2026},
  note={Preprint}
}

License

MIT