An Application of Modular Hyperbolas to Quadratic Residues

نویسندگان

  • Mizan R. Khan
  • Richard Magner
چکیده

For a prime p > 2 let Zp be the group of invertible elements modulo p, and let Hp denote the modular hyperbola xy ≡ 1 (mod p) where x, y ∈ Z. Following [1] we define Hp = Hp ∩ [1, p− 1], that is, Hp = {(x, y) ∈ Z : xy ≡ 1 (mod p), 1 ≤ x, y ≤ p− 1}. We note that the lines l1 : y = x and l2 : y + x = p are lines of symmetry of Hp. In this note we use these two symmetries to prove the following basic fact about quadratic residues.

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عنوان ژورنال:
  • The American Mathematical Monthly

دوره 122  شماره 

صفحات  -

تاریخ انتشار 2015