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

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 $\rho=0.9949$ 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.