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This article is cited in 19 scientific papers (total in 19 papers)
On the number of solutions of the Diophantine equation of Frobenius
S. Sertöz
Abstract:
We consider a linear Diophantine equation of the form
$$
x_1 a_1+\ldots+x_n a_n = N,
$$
where $n$ is a fixed integer greater than one, $0<a_1<\ldots<a_n$ is a fixed set
of integers such that $(a_1,\ldots,a_n)=1$. We denote by $f(N)$ the number of solutions
in non-negative integers. It is well known that $f(x)=P(x)+\Delta(x)$, where
$P(x)$ is a polynomial in $x$ of degree $n-1$ and $\Delta(x)$ is a periodic function
with period $a_1\ldots a_n$. We apply an elementary approach to the problem
of calculating $\Delta(x)$, and utilize roots of unity arguments in
constructing this periodic function.
For $f(N)$, an explicit expression is obtained for arbitrary $n$; this expression
includes complicated sums containing the roots of unity. In the case $n=2$,
this approach leads to a computable explicit expression for $f(x)$.
We note that previously the expression for $\Delta(x)$ has not been known.
Received: 21.10.1996
Citation:
S. Sertöz, “On the number of solutions of the Diophantine equation of Frobenius”, Diskr. Mat., 10:2 (1998), 62–71; Discrete Math. Appl., 8:2 (1998), 153–162
Linking options:
https://www.mathnet.ru/eng/dm428https://doi.org/10.4213/dm428 https://www.mathnet.ru/eng/dm/v10/i2/p62
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