On a problem of Berzsenyi regarding the \(\gcd\) of polynomial expressions (Q1773049)

In 1995, \textit{G. Berzsenyi} [Maximizing the greatest, Quantum 1995, No. 3, 39 (1995)] defined \[ G(m,k) = \max\{\gcd((n+1)^{m} + k, n^{m}+k) \mid n\in \mathbb{N}\}, \] and proved that \(G(2,k)=| 4k+1| \). Here the authors obtain the following explicit formulas for \(G(3,k), G(4,k)\) and \(G(5,k)\) \[ \begin{aligned} G(3,k) &= \begin{cases} 27k^2 +1 & \text{if }k\equiv 0 \pmod{2},\\ (27k^2 +1)/4 & \text{if } k\equiv 1 \pmod{2}, \end{cases}\\ G(5,k)&= \begin{cases} (3125k^4 + 625k^2 + 1)/11 & \text{if } k\equiv \pm 1\pmod{11},\\ 3125k^4 + 625k^2 + 1 & \text{otherwise} \end{cases}\\ \text{and} G(4,k) &= \frac{p_{1}^{\alpha_{1}}\ldots p_{r}^{\alpha_{r}}| 16k+1| }{5^{\varepsilon(k)}}, \quad\text{where } \varepsilon(k)= \begin{cases} 1 & \text{if } k\equiv 1 \pmod{2},\\ 0 & \text{otherwise} \end{cases} \end{aligned} \] and the prime factorization of \(| 4k-1| \) is \(p_{1}^{\alpha_{1}}\ldots p_{r}^{\alpha_{r}}q_{1}^{\beta_{1}}\ldots q_{s}^{\beta_{s}}\), and the \(p_{j}\)'s are all the primes congruent to 1 modulo 4 in this factorization.