Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture

We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by $(n,x,y,z,w)=(2,5,1,1,2)$, $(3,3,2,1,1)$. Our proof of this result for each $n \ge 4$ provides an explicit modulus $M$ such that the corresponding equation has no solution already modulo $M$. This contributes to a classical problem posed by T. Skolem in 1930's on a local-global principle on purely exponential Diophantine equations.