open problem

Goldbach's Conjecture

Every even integer greater than 2 is the sum of two primes. Verified to astronomical bounds, never proven.

10 machine-reviewed papers whose extracted claims name this problem. Matching is literal, over the claim graph, so every listing traces to a sentence in a review.

This is the review hub: papers that name the problem, with the machine's verdict on each. The claims shelf lists the individual claims themselves, formal (Lean) or stated.

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CONDITIONAL Goldbach Representations in Quantum Optics and Metrology
A random beam splitter maps Goldbach representations onto photon-counting outcomes with exact probability |G(2n)|/(2n+1); the associated metrology states scale as the standard quantum limit and as the Heisenberg limit.

arxiv:2610.38675 · quant-ph · 2026-10
ACCEPT Goldbach and Alexandroff Shake Hands
The strong Goldbach conjecture is equivalent to the Goldbach relational space being connected and not satisfying the wR0 separation axiom.

arxiv:2610.25573 · math.GN · 2026-10
CONDITIONAL Structure-Driven Methodology: An Emerging Cross-Domain Paradigm
The practice of identifying and exploiting a problem's intrinsic structure before optimizing is proposed as a cross-domain complement to two centuries of objective-driven inverse-problem solving.

arxiv:2610.02698 · math.HO · 2026-10
REJECT Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions
The paper derives a claimed e^{-N^alpha} upper bound on the probability of Goldbach counterexamples and conjectures a generalized Goldbach property for prime-like sets.

arxiv:2504.14353 · math.NT · 2025-04
CONDITIONAL S-invariant and S-multinvariant functions and some symmetry groups of algebraic sieves
For a dihedral action on Z_{2n}, all S-invariant permutations form the affine group Aff(Z_{2n}), and the Goldbach sieve symmetry group G_N restates the Goldbach conjecture as G_N not equal to Aff(Z_N).

arxiv:2411.17168 · math.GR · 2024-11
CONDITIONAL Bj\"orck Sequences: Extension to Arbitrary Lengths, Correlation Analysis, and Applications to Wireless Systems
A Goldbach-based construction extends Björck CAZAC sequences to arbitrary lengths while preserving cyclic-shift orthogonality and improving high-Doppler robustness over Zadoff-Chu sequences.

arxiv:2506.00706 · eess.SP · 2025-05
UNVERDICTED The Goldbach Conjecture as an Informational Economy Principle: A Heuristic Framework from Computational Physics
The paper frames the Goldbach Conjecture as a consequence of an empirically inferred 'informational economy' among prime pairs, without providing a proof.

arxiv:2508.18273 · physics.soc-ph · 2025-07
CONDITIONAL An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime
Every even number up to a trillion is shown by exhaustive computation to be a prime plus a twin prime, and a proof shows the twin member's orientation is forced by the residue of n mod 3.

arxiv:2608.02381 · math.NT · 2026-08
REJECT Introducing and Applying S.C.E Model Under Dusart's Inequality to Prove Goldbach's Strong Conjecture for 74 Typical Structures out of All 75 Structural Types of Even Number
Introduces S.C.E model categorizing even numbers into 75 structures and claims relative proof of Goldbach's conjecture for 74 structures using Dusart's inequality.

arxiv:1909.13230 · math.NT · 2019-09
UNVERDICTED Studies in Additive Number Theory by Circles of Partition
Introduces circles of partition to prove that large n can be summed from any dense set H (density >1/2) and claims asymptotic proofs of Goldbach and Lemoine conjectures.

arxiv:2012.01329 · math.GM · 2020-12