Quadratic residues of certain types

摘要

The main purpose of the paper is to show that if p is a prime different from 2, 3, 5, 7, 13, 37, then there exists a prime number q smaller than p, q equivalent to 1 (mod 4), which is a quadratic residue modulo p. Also, it is shown that if p is a prime number which is not 2, 3, 5, 7,17, then there exists a prime number q equivalent to 3 (mod 4), q < p, which is a quadratic residue modulo p.