AI系统Theo Conjecture近日破解了一个困扰数学界35年的图论难题。[1]这一猜想源于1987年Siemion Fajtlowicz的Graffiti程序提出的Conjecture 448，[1]涉及Havel-Hakimi残基的增长规律。[1]
Paul Erdős证明了残基增长速率至少为n / log n，其中常数为ζ(2) − 1 ≈ 0.644934。[1]随后William Staton假设这个下界就是精确答案，[1]但这一预测在过去40年间始终未获证明。[1]Theo Conjecture通过与人类数学家的协作循环，利用大语言模型、组合推理引擎和精确计算，[1]最终得出了完整的渐近展开式：R(Gₙ) = c₀ · n/log n + (c₀ − A) · n/log²n + O(n/log³n)。[1]其中A = 0.3201986326...，这一二阶项系数此前未被任何人预测。[1]该系数定义为求和公式Σ(k=2 to ∞) log k / [k²(k−1)]。[1]
在实际验证中，该系统在n ≤ 10,000的连续测试中发现，残基与向上舍入的Caro-Wei和的差值不超过2，[1]进一步验证了理论结果的准确性。
An AI system called Theo Conjecture has solved a longstanding mathematical problem originating from Paul Erdős's work, proving a hypothesis about graph residues that remained unproven for decades [1]. The system discovered a previously unpredicted second-order term in the asymptotic expansion, revealing a formula that connects to the Riemann zeta function [1].
The conjecture traces back to 1987, when Siemion Fajtlowicz's Graffiti program generated Conjecture 448 [1]. Erdős established that the growth rate of Havel-Hakimi residues is at least n / log n with a constant of ζ(2) − 1 ≈ 0.644934 [1]. William Staton subsequently conjectured that this lower bound represented the exact answer, but the proof eluded mathematicians for 40 years [1].
Theo Conjecture's complete formula yields: R(Gₙ) = c₀ · n/log n + (c₀ − A) · n/log²n + O(n/log³n), where A = 0.3201986326... [1]. The newly identified second-order coefficient is defined as A = Σ(k=2 to ∞) log k / [k²(k−1)] [1]. Through collaboration between the AI system and human mathematicians, employing large language models, combinatorial reasoning engines, and precise computation, researchers validated their findings across consecutive tests up to n ≤ 10,000, where residues differed from the rounded Caro-Wei sum by no more than 2 [1].