On the solution of a system of linear Diophantine equations in prime numbers

In the paper, it is proved that the system of linear Diophantine equations consisting of s equations with m unknowns (where s < m ≤ 2s) is solvable in prime numbers with some exceptions. A lower bound for the number of solutions of the system under consideration is also obtained for the exceptional set. The obtained results complement the corresponding results of Wu Fang, M. C. Liu, K. M. Tsang and I. Allakov