On the least pair of consecutive quadratic non-residues
The author shows that for any sufficiently large prime p there is a pair of consecutive quadratic nonresidues modulo p, say n and \(n+1\), with \(n=O(p^{\theta +\epsilon})\) for \(\theta =1/4\sqrt{e}\). This improves an earlier estimate of P.D.T.A. Elliott. Like Elliot's theorem the proof uses the reviewer's character sum estimate. The author's major new tool is the following: For every \(\epsilon\in (0,1]\) there exist positive integers \(N_ 0(\epsilon)\) and \(k_ 0(\epsilon)\) with the property that for any \(N\geq N_ 0(\epsilon)\), and for any set A of positive integers satisfying \[ \sum_{n\leq N, n\equiv \ell (mod k), n\in A}1\geq \epsilon N/k\quad (1\leq k\leq k_ 0(\epsilon),\quad 0\leq \lambda \leq k-1) \] and \(\{\) n/d: \(n\in A\), d \(| n\), \(d\leq k_ 0(\epsilon)\}\subset A\), the set A contains a pair of consecutive integers \(\leq N\).
pair of consecutive quadratic nonresidues
scientific article; zbMATH DE number 4000015