On consecutive k-th power residues
It is known that for fixed \(k\geq 2\) and every sufficiently large prime p there exists an integer r such that r and \(r+1\) are both k-th power residues modulo p. It has been conjectured that the least positive r with this property is bounded in terms of k only. We prove this conjecture in the case k is a prime, and a weaker result in the general case. The main step in the proof consists in showing that for any completely multiplicative function f, whose values are k-th roots of unity, the equation \(f(n)=f(n+1)\), and if k is prime the stronger equation \(f(n)=f(n+1)=1\), has a solution \(n\leq c_ k\), where \(c_ k\) depends only on k. This is achieved by a combinatorial argument.
scientific article; zbMATH DE number 3952879