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

Proofs: 1

Propositions: 1


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998