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On the number of representations of an integer by a linear form. (English) Zbl 1108.11026
Let $$a_1,a_2,\dots,a_k$$ be positive integers with $$\gcd (a_1,a_2,\dots,a_k)=1$$ and $$N$$ be a non-negative integer. Let $$r(N)$$ denote the number of non-negative solutions to the equation $$x_1 a_1 + x_2 a_2 + \dots + x_k a_k = N$$. $$r(N)$$ is related to the Frobenius number, which is the largest integer $$N$$ such that $$r(N)=0$$. In the case of $$k=2$$, $$r(N)$$ is given by a closed formula by T. Popoviciu, [Acad. Repub. Pop. Romane Fil. Cluj Stud. Cerc. Stiint. 4, 7–58 (1953; Zbl 0053.35805)]. In this paper, the authors derive explicit formulas for $$r(N)$$ in the case of $$k=3$$ up to a constant. Their results work well even in many cases where $$a_1, a_2, a_3$$ are not pairwise coprime. In special, the case where $$\gcd(a_1, a_2)\cdot\gcd(a_1, a_3)\cdot\gcd(a_2, a_3)=\text{ lcm}(a_1, a_2, a_3)$$ is studied in more detail.
##### MSC:
 11D04 Linear Diophantine equations
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