On the exponential Diophantine equation $\left(P_{n}^{(k)}\right)^{x}+\left(P_{r}^{(k)}\right)^{x}=P_{m}^{(k)}$

Abstract

The Pell numbers satisfy $P_n^2+P_{n+1}^2=P_{2n+1}$, and Bravo, Luca and Pomeo recently found all the cases in which the sum of the $x$-th powers of two consecutive $k$-generalized Pell numbers is again a $k$-generalized Pell number. We ask what happens when the two indices are arbitrary, that is, we solve $\big(P_n^{(k)}\big)^x+\big(P_r^{(k)}\big)^x=P_m^{(k)}$ in positive integers $k\ge2$, $n\ge r$, $m$ and $x$. The answer is that nothing new appears: apart from $1^x+1^x=2$, every solution has consecutive indices. The proof combines linear forms in logarithms with an auxiliary Fibonacci equation which does not depend on $k$, and, for large $k$, with the fact that the dominant root of the sequence is very close to $φ^2$. For the reduction step, the method of Dujella and Pethő fails for large indices and has to be replaced by a criterion of Legendre type.