Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.7

Counting Integers Representable as Images of Polynomials Modulo n

Fabián Arias
Facultad de Ciencias Básicas
Universidad Tecnológica de Bolívar

Jerson Borja and Luis Rubio
Departamento de Matemáticas y Estadística
Universidad de Córdoba


Given a polynomial f(x1,x2,...,xt) in t variables with integer coefficients and a positive integer n, let α(n) be the number of integers 0 ≤ a < n such that the polynomial congruence f(x1,x2, ...,xt) ≡ a (mod n) is solvable. We describe a method that allows us to determine the function α associated with polynomials of the form c1xk1 + c2xk2 + ··· + ctxkt. Then, we apply this method to polynomials that involve sums and differences of squares, mainly to the polynomials x2 + y2, x2y2, and x2 + y2 + z2.

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Received February 21 2019; revised version received August 13 2019; September 11 2019. Published in Journal of Integer Sequences, September 23 2019.

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