**Proposition**: Diophantine Equations of Congruences

Let $m > 1$ be a positive integer and let $f(x_1,\ldots,x_r)=0$ be a Diophantine equation. If integers $a_1,\ldots,a_r$ solving this equation^{1} exist, then the congruences $a_1(m),\ldots,a_r(m)$ solve also the **Diophantine equation of congruences modulo** $m$, i.e. we have

$$(f(a_1,\ldots,a_r))(m)\equiv f(a_1(m),\ldots,a_r(m))\equiv 0(m).$$

^{1} i.e. by setting $x_r=a_1,\ldots,x_r=a_r.$ Note that a solution does not have to exist, for instance $x^3+y^3=z^3$ has no integer solutions.

| | | | | created: 2019-04-13 15:42:11 | modified: 2019-04-19 07:04:01 | by: *bookofproofs* | references: [1272], [8152]

## 1.**Proof**: *(related to "Diophantine Equations of Congruences")*

## 2.**Corollary**: Diophantine Equations of Congruences Have a Finite Number Of Solutions

[8152] **Jones G., Jones M.**: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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