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 equation1 exist, then the congruences $a_1(m),\ldots,a_r(m)$ solve also the Diophantine equation of congruences modulo $m$, i.e. we have $$(f(a_1,\ldots,a_r))(m)\equiv f(a_1(m),\ldots,a_r(m))\equiv 0(m).$$

Proofs: 1 Corollaries: 1

Corollaries: 1
Proofs: 2 3 4
Propositions: 5 6


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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

Footnotes


  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.