Prime Producing Quadratic Polynomials and Class Number One or Two |
| |
Authors: | Email author" target="_blank">Anitha?SrinivasanEmail author |
| |
Institution: | (1) Departamento de Matematicas, Universidad de Puerto-Rico, Humacao, Puerto Rico, 00791;(2) Present address: Department of Mathematics, Indian Institute of Technology, Powai, Mumbai, India, 400 076 |
| |
Abstract: | Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let
. Assume that
is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if
, then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If
then h< 2 or h = 4. If
, then h≤ 2, h = 4 or h = 2t−2, where t is the number of primes dividing d. In the case when h = 2t−2 we have
, where φ = 2 or 4.
2000 Mathematics Subject Classification: Primary–11R29 |
| |
Keywords: | class number binary quadratic forms quadratic field prime producing polynomials |
本文献已被 SpringerLink 等数据库收录! |
|