SUMS OF POLYNOMIAL TYPE EXCEPTIONAL UNITS MODULO n

Let f (x) is an element of Z[x] be a nonconstant polynomial. Let n >= 1, k >= 2 and c be integers. An integer a is called an f-exunit in the ring Z(n) of residue classes modulo n if gcd(f(a), n) = 1. We use the principle of cross-classification to derive an explicit formula for the number N-k,N-f,N-c(n) of solutions (x(1),..., x(k)) of the congruence x(1) + ... + x(k) c (mod n) with all x(i) being f-exunits in the ring Z(n). This extends a recent result of Anand et al. ['On a question off-exunits in Z/nZ', Arch. Math. (Basel) 116 (2021), 403-409]. We derive a more explicit formula for N-k,N-f,N-c (n) when f (x) is linear or quadratic.