We test Hardy and Littlewood's generalization (GGC) of Goldbach's and Lemoine's conjectures. According to GGC, for relatively prime positive integers m₁ and m₂, every sufficiently large integer n satisfying certain simple congruence criteria can be expressed as n = m₁ p + m₂ q for some primes p and q. We check GGC up to 10¹² for all (up to 10¹³ for some) relatively prime coefficients m₁, m₂ ≤ 40 and present the largest counterexamples that cannot be obtained in this form. We verify Lemoine's conjecture up to a new record of 10¹³. We compare the running times of four natural verification algorithms for all relatively prime m₁ ≤ m₂ ≤ 40. The algorithms seek to find either the p- or the q-minimal (m₁, m₂)-partitions of all numbers tested, by either a descending or an ascending search for the prime to be maximized or minimized, respectively, in the partitions. For all m₁, m₂ descending searches were faster than ascending ones. We provide a heuristic explanation. The relative speed of ascending [descending] searches for the p- and the q-minimal partitions, respectively, varied by m₁, m₂. Using the average of p^*_{m1, m2}(n)—the minimal p in all (m₁, m₂)-partitions of n—up to a sufficiently large threshold, we introduce two functions of m₁, m₂, which may help predict these rankings. Our predictions correspond well with actual rankings, and could inform new verification efforts. Numerical data are presented, including average and maximum values of p^*_{m1, m2}(n) up to 10⁹.