Bounded Gateway Parameters for the Erdos-Straus Conjecture

Gateway Decompositions via Divisors of N^2

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 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.

Separately, the paper proves unconditionally that the residual class is negligible - it has relative density zero among the primes, at the rate O((log X)^{-1/2}) - so a single value A = 3 suffices for a set of primes of density one. See Density-One Theorems.

Theorem. For every prime p <= 10^11 with 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.

Write A(p) for the least prime A = 3 (mod 4) whose gateway succeeds at p = 1 (mod 4). Since 3 is the least prime = 3 (mod 4), A(p) >= 3 always. Paper §5 proves, unconditionally:

Theorem 5.4. #{p <= X : p = 1 (mod 4), A(p) > 3} << X (log X)^{-3/2}.

So A(p) = 3 for a set of primes of density one, and the residual class of §4 has relative density O((log X)^{-1/2}) among the primes. The value 3 is optimal - there is no smaller gateway to shrink to.

Theorem 5.6. For each fixed T, with k(T) = #{A <= T : A prime, A = 3 (mod 4)},

#{p <= X : p = 1 (mod 4), A(p) > T}  <<_T  X (log X)^{-1-k(T)/2}

T = 3 recovers Theorem 5.4; T = 7 gives O(X (log X)^{-2}).

Both are elementary Selberg upper-bound sieves applied to the integers n <= x, with p = 4n-3 and "4n-3 has no prime factor <= z" used as a majorant for "4n-3 is prime". That makes every remainder term elementary integer-counting, so no equidistribution theorem is used anywhere - no Bombieri-Vinogradov, no large sieve, no GRH. The sieve dimension is 1 + k/2: one from the primality majorant, one half per gateway. Only upper bounds are proved; no matching lower bound is claimed.

These say nothing about A(p) being bounded for every prime (that is the open bounded-A conjecture), and nothing about the Erdos-Straus conjecture itself, which is verified far beyond any height relevant here.

Correction. Earlier versions of the paper claimed the residual class had relative density zero on the strength of a Landau-type count of integers whose prime factors all lie in 1 (mod 3) (~ c x/sqrt(log x)). That inference is invalid: dividing by pi(X) ~ X/log X gives c sqrt(log X) -> infinity, and the bound in fact exceeds pi(X) throughout the range of the paper. The conclusion was right; the argument was not. Theorem 5.4 is what establishes it. See paper Remark 5.7.

The maximum A grows extremely slowly with scale, over the range formally verified in this repository (every prime independently cross-checked by a separate implementation, per Verification):

The record-setting prime in this range 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 flatness does not, by itself, support the bounded-A conjecture. A distributional analysis of the tail of A(p) over this range (see the paper, §7.1, and the companion analysis) shows two candidate growth models - one giving an absolutely bounded maximum, the other unbounded but extremely slow - that fit these six data points identically. A single decade of flatness cannot distinguish them.

Consistent with that, an exploratory search past 10^11 (same validated engine as above; every witness with A >= 100 - all 19,540 of them, not a sample - has since been independently re-derived, but the classification of the ordinary small-A primes has not, so still reported separately from the verified table) has since found two further records, at two separate jumps, before the search stopped at 1.212x10^12 with no further record:

Both record witnesses were independently re-derived from scratch by a separate reviewer, and - since - every witness with A >= 100 in the range has been independently re-confirmed the same way: primality, that the divisor genuinely divides N_A^2, and an exhaustive check that every smaller gateway fails. Zero discrepancies across 19,540 witnesses (full results). The two jumps are different sizes (+8, then +112) - together +120 over the interval, comparable to the single 239 -> 359 jump, but as two separate events rather than one. Either way, the occasional-large-jump pattern from the verified range continues past 10^11 rather than flattening further.

None of this refutes the bounded-A conjecture - 479 is a modest value, and the flat-vs-growing question remains genuinely open - but the flatness argument for it should be retired. The conjecture stands on its own as an open question, not as one the 10^10 -> 10^11 data supports. If true, the full Erdos-Straus conjecture follows.

Closing the remaining gap - independently classifying every ordinary prime, not just the large-A witnesses - was costed and deliberately not done. An independent implementation classifies primes at 13,000-19,000/second depending on scale (calibration), so the 33,489,857,205 primes in [10^11, 10^12] alone would cost roughly 1-3 days of wall time even parallelised. The value of max A is already exhaustively settled by the witness check above, and a bug severe enough to silently misclassify billions of easy primes would very likely also have corrupted the gateway search already re-verified - so the additional confidence didn't justify several days of compute. Reported at this standard deliberately.

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: The unconditional density-one theorems above, the identification of Case B with the failure of gateway 3 (Lemma 5.3), proof sketches, and the correction to the earlier density argument (Remark 5.7)
• Section 6: 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 7: 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 4.4 distinguishes them with explicit examples.

• unconditional_bound.tex - An earlier, weaker 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, a half-dimensional Selberg sieve, and a Bombieri-Vinogradov level of distribution. 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. Superseded on the density bound by Theorem 5.6 above (T = 7), which bounds a larger set by O(X (log X)^{-2}) and needs no Bombieri-Vinogradov input; the sharpness and barrier discussion stands.

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 6.4):

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.

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

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.

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 6 of the paper documents the correction.

@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}
}

MIT