NOTES ON PRIMES P 1 mod D AND AP 1/D 1 mod P

摘要

Let d > 0 be a squarefree integer and a be an integer, which is not 1 nor a square. Let P(a,d)(x) be the number of primes p ? x such that p 1 mod d and a(p1)/d 1 mod p. Numerical data indicate that the function as approximately equal to a constant multiple of ?(x)/(d'(d)) for suciently large x, where ?(x) is the number of primes up to x and '(d) is the Euler-' function. The involved constant multiple depends on both a and d. In this paper we obtain an average order of the function and explore some properties of the primes counted by the function.