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

Facultad de Ciencias Básicas

Universidad Tecnológica de Bolívar

Colombia

Jerson Borja and Luis Rubio

Departamento de Matemáticas y Estadística

Universidad de Córdoba

Colombia

**Abstract:**

Given a polynomial
*f*(*x*_{1},*x*_{2},...,*x*_{t})
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*(*x*_{1},*x*_{2},
...,*x*_{t}) ≡ *a* (mod *n*)
is solvable.
We describe a method that allows us to determine the
function α associated with polynomials of the
form *c*_{1}*x*^{k1} +
*c*_{2}*x*^{k2}
+ ··· +
*c*_{t}*x*^{kt}.
Then, we apply this method to polynomials
that involve sums and differences of squares,
mainly to the polynomials
*x*^{2} + *y*^{2},
*x*^{2} − *y*^{2},
and *x*^{2} + *y*^{2} + *z*^{2}.

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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