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Nested Squares and Distinct-Prime Goldbach Partitions

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.NA — Numerical Analysis

Contributed by Frank Vega ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

The distinct-prime Goldbach conjecture asserts that for every N≥4N\geq 4 there is M∈AN=[1,N−3]M\in A_N=[1,N-3] with N−MN-M and N+MN+M both prime; geometrically, nested squares of side lengths NN and MM have an L-shaped difference of semiprime area (N−M)(N+M)(N-M)(N+M). With CNC_N and DND_N the sets of M∈ANM\in A_N for which N−MN-M, respectively N+MN+M, is prime, the conjecture for 2N2N is CN∩DN≠∅C_N\cap D_N\neq\varnothing. We embed this centered condition in a straddling one: CN∩DNC_N\cap D_N lies in the set ENE_N of half-differences (Q−P)/2∈AN(Q-P)/2\in A_N of primes 2<P<N<Q<2N2<P<N<Q<2N. From Mikawa's theorem on prime pairs with almost all differences we deduce ∣AN∖EN∣≪DN/(log⁡N)D|A_N\setminus E_N|\ll_D N/(\log N)^D for every D>0D>0, and with explicit constants we prove ∣EN∣>N/760|E_N|>N/760 for every N≥3275N\geq 3275. Consequently, for all large NN and all but OD(N/(log⁡N)D)O_D(N/(\log N)^D) primes P<NP<N, the number 2N−P2N-P becomes prime after adding the difference P′−PP'-P of two primes P′<NP'<N; Goldbach's partition is the case P′=PP'=P. In contrast, ∣CN∩DN∣<(N−3)−log⁡2(2N)|C_N\cap D_N|<(N-3)-\log^2(2N) for every N≥34N\geq 34, and covering all but log⁡2(2N)\log^2(2N) values of ANA_N for all large NN would imply the Cram\'er-type bound p′−p<2log⁡2(2p+2)+2p'-p<2\log^2(2p+2)+2. The method also shows that 2N2N is a sum of two distinct primes for more than X/9600X/9600 integers N≤XN\leq X, for every X≥106X\geq 10^6. Computations up to N=106N=10^6 compare the number of distinct-prime partitions with the Hardy--Littlewood prediction and measure the missing values of ENE_N forced by prime gaps. The conjecture itself is not proved here.

Provenance statement

The author used Claude Opus 5.5 to write the experiment scripts experiment.py and large_range.py, and to improve the readability of this manuscript; the author ran all the experiments. After using this tool, the author reviewed and edited the code and the content as needed and takes full responsibility for the content of the publication.

Tools used

Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Frank VegaInitial depositCurrentNov 03, 2026