Sufficient conditions for solvability of linear Diophantine equations and Frobenius numbers

A linear Diophantine equation ∑ᵢ₌₁ⁿaᵢxᵢ=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 [b/M]≥ [n-∑ᵢ₌₁ⁿaᵢ+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-∑ᵢ₌₁ⁿaᵢ or b≥ (n-1)M. The case where ∑ᵢ₌₁ⁿaᵢ≥ (n-2)M+2 is considered closer. It is proved that, in that case, the equation is solvable if b>(n-1)M-∑ᵢ₌₁ⁿaᵢ. If, in addition, r>(n-1)M-∑ᵢ₌₁ⁿaᵢ, then it has Mⁿ⁻¹a₁a₂...aₙCⁿ⁻¹b'+n-1 integer non-negative solutions; it is also shown that if b≥ M and r≤ (n-1)M-∑ᵢ₌₁ⁿaᵢ, then the number of solutions of the equation is greater than or equal to Mⁿ⁻¹a₁a₂...aₙCⁿ⁻¹b'+n-2; here b'=[b/M] (and Cᵏₘ 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).