Prime Producing Quadratic Polynomials and Class Number One or Two

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 $f_p(x)=\vert p{x}^{2} + px + \frac{p-q}{4}\vert$. Assume that $f_p(x)$ 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 $W=\frac{1}{2}(\sqrt{\frac{d}{5}}-1)$, 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 $W=\frac{\sqrt{d}}{4}-\frac{1}{2}$ then h< 2 or h = 4. If $W=\frac{1}{2}(\sqrt{\frac{d}{5}}-1)$, then h≤ 2, h = 4 or h = 2^t−2, where t is the number of primes dividing d. In the case when h = 2^t−2 we have $d=p^{2}\phi^{2}\pm p$, where φ = 2 or 4.