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Predicting the size ranking of minimal primes in the generalised Goldbach partitions

Zsófia Juhász, Máté Bartalos

TL;DR

The paper investigates the size ranking of minimal primes in the generalised Goldbach partitions by introducing a rank predictor $R_{m_1,m_2}$ that depends only on $m_2$ and the prime factors of $m_1$. Through extensive computation up to $n \le 10^9$ for coprime pairs $(m_1,m_2)$ with $1 \le m_1 \neq m_2 \le 40$, the authors demonstrate an exceptionally strong Spearman correlation between $R_{m_1,m_2}$ and both the average and maximum of $p^*_{m_1,m_2}(n)$, suggesting a robust, factor-based structure in these partitions. They provide theoretical heuristics via the Chinese Remainder Theorem and the Prime Number Theorem for Arithmetic Progressions, and they observe predictive trends such as larger averages for $(m_1,m_2)$ with $m_1 < m_2$, and near invariance of averages when $m_1$ shares the same prime divisors. The work also links $p^*_{m_1,m_2}(n)$ to the practical running times of GGC verification algorithms, proposing a runtime proxy $g_L$ and showing that large-prime strategies often outperform small-prime strategies in practice. Overall, the findings offer a principled, arithmetic-structure-based forecast of partition sizes and algorithmic workloads in the generalized Goldbach setting.

Abstract

A scarcely known generalization of Goldbach's conjecture introduced by Hardy and Littlewood states that for every pair of (relatively prime) positive integers m1 and m2, every sufficiently large integer n satisfying certain simple congruence criteria can be $(m_1,m_2)$-partitioned as $n = m_1p+m_2q$ for some primes $p$ and $q$. While the size of the minimal prime in the Goldbach partitions of even numbers has received prior attention, we extend this investigation to the general case of $(m_1,m_2)$-partitions. This question has a direct implication on the running times of verification algorithms of the generalised Goldbach conjecture. We study the rankings of the pairs $(m_1,m_2)$ according to the sizes of the averages and maxima, respectively, of the minimal $p$ in the $(m_1,m_2)$-partitions of numbers up to large thresholds, and propose a rank-order predicting function depending only on m2 and the prime factors of $m_1$. We computed both the average and the maximum of the minimal prime $p$ in all $(m_1,m_2)$-partitions of integers up to $10^9$, for every pair of relatively prime coefficients $1\leq m1\neq m2\leq 40$. Our function shows very high rank-order correlations with both the empirical averages and maxima of the minimal primes $p$ (Spearman's $ρ=0.9949$ and $0.9958$, respectively). It also correctly predicts trends in the experimental data, for example, that for all relatively prime $1\leq m1<m2\leq 40$, the average minimal $p$ in the $(m_1,m_2)$-partitions of numbers up to $10^9$ exceeds the analogous average for the $(m_2,m_1)$-partitions. We present numerical data, including the average and the maximum of the minimal $p$ in the $(m_1,m_2)$-partitions of numbers up to $10^9$ for each pair $1\leq m1\neq m2\leq 20$ relatively prime, and the resulting size rankings.

Predicting the size ranking of minimal primes in the generalised Goldbach partitions

TL;DR

The paper investigates the size ranking of minimal primes in the generalised Goldbach partitions by introducing a rank predictor that depends only on and the prime factors of . Through extensive computation up to for coprime pairs with , the authors demonstrate an exceptionally strong Spearman correlation between and both the average and maximum of , suggesting a robust, factor-based structure in these partitions. They provide theoretical heuristics via the Chinese Remainder Theorem and the Prime Number Theorem for Arithmetic Progressions, and they observe predictive trends such as larger averages for with , and near invariance of averages when shares the same prime divisors. The work also links to the practical running times of GGC verification algorithms, proposing a runtime proxy and showing that large-prime strategies often outperform small-prime strategies in practice. Overall, the findings offer a principled, arithmetic-structure-based forecast of partition sizes and algorithmic workloads in the generalized Goldbach setting.

Abstract

A scarcely known generalization of Goldbach's conjecture introduced by Hardy and Littlewood states that for every pair of (relatively prime) positive integers m1 and m2, every sufficiently large integer n satisfying certain simple congruence criteria can be -partitioned as for some primes and . While the size of the minimal prime in the Goldbach partitions of even numbers has received prior attention, we extend this investigation to the general case of -partitions. This question has a direct implication on the running times of verification algorithms of the generalised Goldbach conjecture. We study the rankings of the pairs according to the sizes of the averages and maxima, respectively, of the minimal in the -partitions of numbers up to large thresholds, and propose a rank-order predicting function depending only on m2 and the prime factors of . We computed both the average and the maximum of the minimal prime in all -partitions of integers up to , for every pair of relatively prime coefficients . Our function shows very high rank-order correlations with both the empirical averages and maxima of the minimal primes (Spearman's and , respectively). It also correctly predicts trends in the experimental data, for example, that for all relatively prime , the average minimal in the -partitions of numbers up to exceeds the analogous average for the -partitions. We present numerical data, including the average and the maximum of the minimal in the -partitions of numbers up to for each pair relatively prime, and the resulting size rankings.
Paper Structure (6 sections, 1 theorem, 12 equations, 2 figures, 2 tables)

This paper contains 6 sections, 1 theorem, 12 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function $R_{m_1, m_2}$: Comparing the average and the maximum $p^*_{m_1, m_2}(n)$ values, respectively, for different pairs $m_1, m_2$
  4. Connections between the magnitude of $p^*_{m_1, m_2}(n)$ and the running times of verifying algorithms of GGC
  5. Conclusion and closing remarks
  6. Tables of data

Key Result

Proposition 3.1

Let $m_1, m_2$ and $n$ be pairwise relatively prime positive integers satisfying the conditions of GGC. If $m_2\neq 1$ then denote the canonical decomposition of $m_2$ by $m_2=r_1^{\alpha_1}\cdot\ldots \cdot r_k^{\alpha_k}$. For every prime $p<\frac{n}{m_1}$ and $q$ the following two statements are

Figures (2)

  • Figure 1: Subfigures \ref{['subfig1']} and \ref{['subfig2']} plot the ascending ranking of all pairs of relatively prime $1\leq m_1 \neq m_2 \leq 40$ by $R_{m_1, m_2}$ on the vertical axis against their ascending ranking by $\overline{p^*}_{m_1, m_2}$ up to $10^9$ and against their ascending ranking by the maximum of $p^*_{m_1, m_2}$ up to $10^9$, respectively. Subfigure \ref{['subfig3']} displays the ascending ranking of the same pairs ${m_1, m_2}$ by the maximum of $p^*_{m_1, m_2}$ up to $10^9$ against their ascending ranking by $\overline{p^*}_{m_1, m_2}$ up to $10^9$.
  • Figure 2: $R_{m_1, m_2}$ (vertical axis) plotted against $\overline{p^*}_{m_1, m_2}$ up to $10^9$ (horizontal axis) for every pair of relatively prime coefficients $1\leq m_1, m_2 \leq 40$.

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof