Primitive points on a modular hyperbola

Igor E. Shparlinski

Research output: Contribution to journalArticlepeer-review


For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U]x[1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) = 1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are “visible” from the origin.
Original languageEnglish
Pages (from-to)193-200
Number of pages8
JournalBulletin of the Polish Academy of Sciences. Mathematics
Issue number3
Publication statusPublished - 2006


  • congruences
  • primitive roots
  • irregularities of distribution
  • estimates on character sums


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