Growth of the largest strong Goldbach number $L_n$: a quantitative prime-indexed law implying the binary Goldbach conjecture

The binary Goldbach conjecture — every even integer greater than 2 is a sum of two primes — is equivalent, by an elementary argument of Pingyuan Zhou (2017), to the statement that the sequence $L_n$ tends to infinity, where $L_n$ is the largest even integer such that every even integer from 6 to $L_n$ is a sum of two odd primes among the first $n$ primes. Zhou's program poses the open problem of establishing a quantitative, prime-indexed law relating $n$ and $L_n$ — most sharply $A(P_n)=L_n-P_n\approx n\log n$ with $P_n$ the $n$-th prime, alongside the limit relations $L_n/(2P_n)\to 1$ and $L_n/(P_n+n\log n)\to 1$ — any provable unbounded version of which yields a proof of Goldbach's conjecture. The framework remains open: computations support the conjectured laws (the 2024 two-sided bounds are verified for $20542\le n\le 4\cdot 10^8$, with $L_n$ itself tabulated for the first $4\cdot 10^6$ primes), and a proven sufficient bound $L_n\ge P_n-n\log n$ is known to imply the conjecture, but no unconditional lower bound on $L_n$ tending to infinity is known, and the problem is exactly as hard as Goldbach itself.

Background

The binary (strong) Goldbach conjecture, stated in the 1742 correspondence between Christian Goldbach and Leonhard Euler, asserts that every even integer greater than 2 is the sum of two primes. It remains open. The best partial results are of three kinds. First, computational verification: every even integer up to $4\cdot 10^{18}$ has a two-prime representation (Oliveira e Silva, Herzog, and Pardi, 2014). Second, almost-all and density results: writing $E(x)$ for the exceptional set — the number of even integers up to $x$ that are not sums of two odd primes — Montgomery and Vaughan (1975) proved a power saving $E(x)\ll x^{1-\delta}$, and Pintz (2018) improved this to $E(x)<x^{0.72}$; such bounds show failures are sparse but permit infinitely many of them. Third, approximate substitutes: Chen's theorem (1973) gives every sufficiently large even integer as a prime plus a number that is either prime or a product of two primes, and Helfgott (2013) proved the ternary Goldbach conjecture (every odd integer greater than 5 is a sum of three primes). None of these covers every even integer from the start.

Zhou (2017) introduced a constructive, prime-indexed reframing. Let $P_n$ denote the $n$-th prime. A Goldbach number generated by $P_n$ is an even integer of the form $p+q$ where $p,q$ are odd primes not exceeding $P_n$; a strong Goldbach number generated by $P_n$ is such a number $G_n$ with the additional property that every even integer from 6 up to $G_n$ is also a Goldbach number generated by $P_n$; and $L_n$ is the largest strong Goldbach number generated by $P_n$, so that the strong Goldbach numbers generated by $P_n$ are exactly the initial block $\{6,8,\dots,L_n\}$ and $L_n+2$ fails. Basic facts: $L_n\le 2P_n$ (the largest possible sum is $P_n+P_n$); the blocks nest, $\{6,\dots,L_n\}\subseteq\{6,\dots,L_{n+1}\}$, so $L_n$ is nondecreasing and its distribution is a step-shaped curve; and Zhou's Proposition 3.5 (elementary, both directions) states that $L_n\to\infty$ as $n\to\infty$ is equivalent to the Goldbach conjecture. Writing $A(P_n)=L_n-P_n$, Zhou proposed on the basis of computations for $P_n\le 2000$ the quantitative proposition $A(P_n)\approx n\log n$ (Proposition 3.2), which would give $L_n\approx P_n+n\log n$ and hence $L_n\to\infty$. Later work in the same program (Zhou and Ao, 2018; Zhou, 2019, 2024) refined the conjectured law — the limit relations $L_n/(2P_n)\to 1$ and $L_n/(P_n+n\log n)\to 1$, the sufficient bound $L_n\ge P_n-n\log n$, and the two-sided approximate bounds $2n\log n+2n\log\log n-2n<L_n<2n\log n+2n\log\log n$ for $n\ge 20542$ — and verified them numerically as far as $n=4\cdot 10^8$ for the two-sided bounds (with $L_n$ itself tabulated for the first $4\cdot 10^6$ primes), but proved none of them.

Problem Statement

Establish a quantitative, prime-indexed law relating the index $n$ of the $n$-th prime $P_n$ to the largest strong Goldbach number $L_n$ — the largest even integer such that every even integer from 6 to $L_n$ is a sum of two odd primes not exceeding $P_n$ (following Zhou's convention the prime 2 is excluded, so the covered initial block begins at $6=3+3$; since $4=2+2$, this convention is immaterial to the conjecture) — that provably forces $L_n\to\infty$ as $n\to\infty$, and thereby prove the binary Goldbach conjecture (every even integer greater than 2 is a sum of two primes). Admissible precise targets, any one of which resolves the problem, are: (i) Zhou's proposition $A(P_n)=L_n-P_n\approx n\log n$ for all $n$ (in its sharp asymptotic reading, $A(P_n)\sim n\log n$); (ii) either of the limit relations $L_n/(2P_n)\to 1$ or $L_n/(P_n+n\log n)\to 1$; (iii) the sufficient bound $L_n\ge P_n-n\log n$ for all $n\ge 2$; (iv) the two-sided bounds $2n\log n+2n\log\log n-2n<L_n<2n\log n+2n\log\log n$ for all sufficiently large $n$; or (v) any other explicit function $f(n)\to\infty$ together with a proof that $L_n\ge f(n)$ for all sufficiently large $n$. Because $L_n\to\infty$ is equivalent to the Goldbach conjecture, no density, almost-all, or short-interval-existence surrogate substitutes for the required law: the relation must cover every even integer in the initial segments. A single explicit even integer with no representation as a sum of two primes would refute the possibility of any such law.

References

1. Pingyuan Zhou (2017), 'Strong Goldbach Number in Goldbach's Problem', Journal of Mathematics Research 9(6), 95–105. DOI: 10.5539/jmr.v9n6p95, https://doi.org/10.5539/jmr.v9n6p95
2. Pingyuan Zhou and Rong Ao (2018), 'Distribution of the Largest Strong Goldbach Numbers Generated by Primes', Journal of Mathematics Research 10(5), 1–8. DOI: 10.5539/jmr.v10n5p1, https://doi.org/10.5539/jmr.v10n5p1
3. Pingyuan Zhou (2019), 'A weak method to come close to solution of Goldbach conjecture', International Mathematical Forum 14(6), 247–252. DOI: 10.12988/imf.2019.9938, https://doi.org/10.12988/imf.2019.9938
4. Pingyuan Zhou (2024), 'Prime Number Theorem and Goldbach Conjecture', Journal of Mathematics Research 16(3), p. 1. DOI: 10.5539/jmr.v16n3p1, https://doi.org/10.5539/jmr.v16n3p1
5. Pingyuan Zhou (2025), 'On the Connections between Goldbach Conjecture and Prime Number Theorem', Advances in Pure Mathematics 15(6), 412–457. DOI: 10.4236/apm.2025.156020, https://doi.org/10.4236/apm.2025.156020
6. H. Montgomery and R. Vaughan (1975), 'The exceptional set of Goldbach's problem', Acta Arithmetica 27, 353–370. DOI: 10.4064/aa-27-1-353-370, https://doi.org/10.4064/aa-27-1-353-370
7. János Pintz (2018), 'A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach's problem', arXiv:1804.09084. https://arxiv.org/abs/1804.09084
8. Gautami Bhowmik and Lasse Grimmelt (2026), 'The exceptional set of the Goldbach problem', arXiv:2607.27282 (survey with new explicit major-arc formulas). https://arxiv.org/abs/2607.27282
9. Jing-Run Chen (1973), 'On the representation of a larger even integer as the sum of a prime and the product of at most two primes', Sci. Sinica 16, 157–176; reprint in Y. Wang (ed.), The Goldbach Conjecture, World Scientific. DOI: 10.1142/9789812776600_0021, https://doi.org/10.1142/9789812776600_0021
10. H. A. Helfgott (2013), 'The ternary Goldbach conjecture is true', arXiv:1312.7748. https://arxiv.org/abs/1312.7748
11. Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi (2014), 'Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸', Mathematics of Computation 83, 2033–2060. DOI: 10.1090/S0025-5718-2013-02787-1, https://doi.org/10.1090/S0025-5718-2013-02787-1
12. Michael Hellus, Anton Rechenauer, and Rolf Waldi (2020), 'Numerical semigroups generated by primes', Semigroup Forum 101, 690–703. DOI: 10.1007/s00233-020-10102-9, https://doi.org/10.1007/s00233-020-10102-9
13. Andrés Chirre and Markus Valås Hagen (2025), 'On Goldbach numbers in short intervals', arXiv:2512.23534. https://arxiv.org/abs/2512.23534