A Diophantine Problem Solved Through Arithmetic Progressions

We investigate a Diophantine equation involving two distinct prime numbers and establish two complementary existence results using arithmetic progressions and elementary congruence arguments combined with classical estimates on primes in arithmetic progressions. First, we prove that there exists a constant (N_0>0) such that, for every even integer (N\geq N_0) and every prime (q<\sqrt N) satisfying (q\nmid N), there exists a prime (p<\sqrt N), with (p\neq q) and (p\nmid N), for which the equation [ pu+qv=N ] admits an integer solution ((u,v)) satisfying (u\not\equiv 1\pmod q). Second, we fix a prime (p<\sqrt N) with (p\nmid N) and study the primes (q<\sqrt N) for which the same equation admits an integer solution satisfying the corresponding congruence condition. We show that the number (R_p(N)) of such primes satisfies [ R_p(N)\sim \pi(\sqrt N) ] as (N) tends to infinity through even integers. The proof relies on the characterization of the forbidden residue class modulo a prime, estimates for primes in arithmetic progressions, and bounds on the number of distinct prime divisors of an integer. These results show that, for sufficiently large even (N), suitable pairs of distinct primes exist and that, when one prime is fixed, the required property holds for asymptotically almost all admissible choices of the other prime.