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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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