Sufficient conditions for solvability of linear Diophantine equations and Frobenius numbers

A linear Diophantine equation ₈=₁^naᵢxᵢ=b is considered, where a₁, a₂,. . . , aₙ are coprime natural numbers, b is an non-negative integer, xᵢ (i=1, 2,. . . , n) are non-negative integers. It is proved that if M -₈=₁^₍aᵢ+rM, then this equation is solvable (here M is the least common multiple of the numbers a₁, a₂,. . . , aₙ; r is the remainder of b modulo M). With the aid of this result, it is shown that the considered equation is solvable if either b nM-₈=₁^naᵢ or b (n-1) M. The case where ₈=₁^naᵢ (n-2) M+2 is considered closer. It is proved that, in that case, the equation is solvable if b> (n-1) M-₈=₁^naᵢ. If, in addition, r> (n-1) M-₈=₁^naᵢ, then it has M^n-1a₁a₂. . . aₙC^n-1₁'+₍-₁ integer non-negative solutions; it is also shown that if b M and r (n-1) M-₈=₁^naᵢ, then the number of solutions of the equation is greater than or equal to M^n-1a₁a₂. . . aₙC^n-1₁'+₍-₂; here b'=M (and C^k₌ denotes the binomial coefficient pmatrixm\\). Moreover, it is proved that if the numbers a₁, a₂,. . . , aₙ are coprime, then Frob (a₁, a₂,. . . , aₙ) c, where c is the smallest among the numbers aᵢaⱼ-aᵢ-aⱼ with (aᵢ, aⱼ) =1 (1 i< j n).