The reducibility of the generalized cyclotomic equations modulo an odd integer (Q1200069)

Let \(p_ 1^{d_ 1},\dots,p_ r^{d_ r}\) be distinct odd prime powers. Set \(q_ i=p_ i^{d_ i}\) and \(n=q_ 1\dots q_ r\). Let \(g_ i\) be a primitive root of \(q_ i\). Define \(g\) uniquely modulo \(n\) by \(g\equiv g_ i\pmod {q_ i}\) (\(i=1,\dots,r\)). Set \(f_ i=p_ i^{d_ i-1}(p_ i-1)\) and \(f=\text{lcm}(f_ 1,\dots,f_ r)\). If \(a\) and \(b\) are integers with \(\text{gcd}(ab,n)=1\) the number \(N(a,b;g;n)\) of solutions \((s,t)\) of \(ag^ s+1\equiv bg^ t\pmod n\), \(0\leq s,t<f\), is called a generalized cyclotomic number. A number of results about generalized cyclotomic numbers are proved.