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

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 $1(8),3(8),5(8),7(8).$

| | | | | created: 2019-04-13 16:07:05 | modified: 2019-04-13 16:31:03 | by: *bookofproofs* | references: [1272], [8152]

## 1.**Proof**: *(related to "Diophantine Equations of Congruences Have a Finite Number Of Solutions")*

(none)

[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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