The authors take a prime power \(q=p^k\) and a positive integer \(d\) of the form \(d\equiv 1\pmod {q-1}\) and study the number of solutions \(x\in \text{GF}(q^2)\) of the equation \((x+1)^d=x^d+1\). They prove various results concerning this number of the following flavor. Write \(d=(q-1)s+1\). If \(\gcd(s,q+1)=1\), and \(\gcd(s-1,q+1)=2\), then the above equation has exactly \(q\) solutions (Theorem 1). The same holds when \(\gcd(s,q+1)=2\) and \(\gcd(s-1,q+1)=1\). On the other hand, if both \(\gcd(s,q+1)>2\) and \(\gcd(s-1,q+1)>2\), then the above equation has a solution outside of \(\text{GF}(q)\) (Theorem 2). In the last section of the paper, the authors use their results to give a simpler proof of a result of \textit{P. Charpin} [``Cyclic codes with few weights and Niho exponents'', J. Comb. Theory, Ser A. 108, No. 2, 247--259 (2004; Zbl 1072.94016)], on the value set of the cross-correlation of two maximal linearly recurrent sequences first studied by \textit{Y. Niho} [Multivalued cross-correlation functions between two maximal linear recursive sequences, Ph.D. thesis, University or Southern California, Los Angeles, CA (1972)]. This value set can be described in terms of the number of solutions of a system of two polynomial equations in \(\text{GF}(q)\), and the authors results apply to deal with this number.