On the p-adic zeros of polynomials with Bernoulli coefficients
If p is an odd prime, let \(\beta_ p(X)=\sum_{k}(B_ k/k)(X^{p-1- k}-1)\), where \(B_ k\) is the kth Bernoulli number and the summation extends over \(1\leq k\leq p-2\). In this paper we prove the following theorem: Let p be an odd prime number and let \(\alpha\) be an integer such that \(\alpha\geq 2\). If \(p>\alpha^{\pi (\alpha)} \log \alpha\), where \(\pi\) (\(\alpha)\) denotes the number of primes \(\leq \alpha\), and \(\alpha^{p-1}\equiv 1 (mod p^ 2)\), then, for some positive constant C(\(\alpha)\), the polynomial \(\beta_ p(X)\) has \(n>C(\alpha) \log \log p\) distinct simple zeros \(x_ i\) in the ring of p-adic integers. Furthermore, for some integral sequence \(0\leq \tau (1)<...<\tau (n)\leq n\), it follows that \(x_ i\equiv \alpha^{\tau (i)} (mod p)\) for all \(1\leq i\leq n\).
scientific article; zbMATH DE number 3897022