The Diophantine Frobenius Problem revisited

Let $k\ge 2$ and $a_1,a_2,\cdots ,a_k$ be positive integers with $$\gcd (a_1,a_2,\cdots ,a_k)=1.$$ It is proved that there exists a positive integer $G_{a_1,a_2,\cdots ,a_k}$ such that every integer $n$ strictly greater than it can be represented as the form $$n=a_1{x}_1+a_2{x}_2+\cdots +a_k{x}_k,(x_1,x_2,\cdots ,x_k\in {\mathbb{Z}}_{\ge 0},\gcd (x_1,x_2,\cdots ,x_k)=1)\mathrm{.}$$ We then investigate the size of $G_{a_1,a_2}$ explicitly. Our result strengthens the primality requirement of $x$’s in the classical Diophantine Frobenius Problem.