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Diskretnaya Matematika, 1998, Volume 10, Issue 2, Pages 62–71
DOI: https://doi.org/10.4213/dm428
(Mi dm428)
 

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
Bibliographic databases:
UDC: 519.1
Language: Russian
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
Citation in format AMSBIB
\Bibitem{Ser98}
\by S.~Sert\"oz
\paper On the number of solutions of the Diophantine equation of Frobenius
\jour Diskr. Mat.
\yr 1998
\vol 10
\issue 2
\pages 62--71
\mathnet{http://mi.mathnet.ru/dm428}
\crossref{https://doi.org/10.4213/dm428}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1673087}
\zmath{https://zbmath.org/?q=an:0976.11015}
\transl
\jour Discrete Math. Appl.
\yr 1998
\vol 8
\issue 2
\pages 153--162
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  • https://www.mathnet.ru/eng/dm/v10/i2/p62
  • This publication is cited in the following 19 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Дискретная математика
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