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

Zsófia Juhász; Máté Bartalos

paper

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

arXiv:2510.21870

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 $n = m_{1} p + m_{2} q$ 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 $1 0^{9}$ , for every pair of relatively prime coefficients $1 \leq m 1 \neq = m 2 \leq 40$ . 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 $1 \leq m 1 < m 2 \leq 40$ , the average minimal $p$ in the $(m_{1}, m_{2})$ -partitions of numbers up to $1 0^{9}$ 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 $1 0^{9}$ for each pair $1 \leq m 1 \neq = m 2 \leq 20$ relatively prime, and the resulting size rankings.