◀ ▲ ▶Branches / Numbertheory / Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions
Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions
(related to Proposition: Diophantine Equations of Congruences)
Let $m > 1$ be a positive integer and let $(f(x_1,\ldots,x_r))(m)=0(m)$ be a Diophantine equation of congruences modulo $m$. Then the number of distinct solutions, i.e. ordered tuples $(a_1(m),\ldots,a_r(m))$ solving this equation
$$f(a_1(m),\ldots,a_r(m))\equiv0(m)$$
is finite.
Example
The equation $x^2(8)1(8)\equiv 0(8)$ (can also be written as $x^2\equiv 1\mod 8$) has only the four solutions $1(8),3(8),5(8),7(8).$
Table of Contents
Proofs: 1
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Propositions: 1
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
 Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998