For a prime p and an absolutely irreducible modulo p polynomial f(U, V) E Z[U, V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.
|Number of pages||7|
|Journal||Bulletin of the Polish Academy of Sciences. Mathematics|
|Publication status||Published - 2007|
- points visible from the origin
- absolutely irreducible polynomial