Symmetric and asymmetric primes
Let \(P\) denote the set of all odd primes and define \(S: P\times P\to \mathbb{N}\) by \[ S(p, q)= \sum_{k=1}^{(p- 1)/2} [qk/ p]. \] The authors call a pair of different odd primes \((p, q)\) symmetric provided that \(S(p, q)= S(q, p)\). A prime that belongs to a symmetric pair is a symmetric prime; otherwise it is called asymmetric. The authors prove that a pair \((p, q)\), \(p\neq q\) is symmetric if and only if \(p-q\) divides \(p-1\). They also give five other characterizations of a similar type (Proposition 2.1). Numerical computations show that of the first 100,000 odd primes about 5/6 of them are symmetric. This suggests that the density of symmetric primes is positive and perhaps near 0.8. The authors prove however that this count is misleading. Theorem 3.1: For all sufficiently large numbers \(x\), the number of symmetric primes up to \(x\) is \(\leq x/(\log x)^{1.027}\). The method of the proof is based on the characterization of symmetric pairs of some sieve estimates. At the end of the paper the authors conjecture that there is some number \(\sigma> 1\) such that the number of symmetric primes up to \(x\) is \(x/ (\log x)^{\sigma+ o(1)}\) as \(x\to \infty\). They give heuristic arguments that \(\sigma= 2- (1+\log \log 2)/ \log 2= 1.08607 \dots\;\).
scientific article; zbMATH DE number 883837