Abstract
The theorems of Bachet–Bézout and Gauss are the cornerstones of elementary methods in number theory. Many applications of these results are studied, in particular Diophantine problems. The section Further Developments investigates the number of integer solutions of certain linear Diophantine equations, i.e. the number of certain restricted partitions of an integer.
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Notes
- 1.
We could also use Fermat’s little theorem. See Theorem 3.15.
- 2.
When k=1, this obviously means \(\mathcal{A}_{1}= \{ 1\}\).
- 3.
See Exercise 11 for another proof of this identity.
- 4.
One can make use of Pick’s formula which states that the number \(\mathcal{N}_{\mathrm{int}}\) inside a convex integer polygon \(\mathcal{P}\) is given by
$$\mathcal{N}_{\mathrm{int}} = \mathrm{area} ( \mathcal{P} ) - \frac {\mathcal{N}_{\partial\mathcal{P}}}{2} + 1 $$where \(\mathcal{N}_{\partial\mathcal{P}}\) is the number of integer points on the edges of the polygon.
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Bordellès, O. (2012). Bézout and Gauss. In: Arithmetic Tales. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4096-2_2
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