A Richaud-Degert type real quadratic field is a number field of the form $$K = \mathbb{Q}(\sqrt{d})$$ where $$d = {(an)}^2 + ka > 0$$ for positive integers $$a, n$$ and $$k \in \{ \pm 1, \pm 2, \pm 4 \}$$, $$-n < k \leq n$$, $$d \neq 5$$ and $$d$$ square free.

The reason to define such strange $$d$$ is that it turns out that the fundamental unit of such real quadratic extensions is small, typically of the order of at most $$d$$. So by Siegel's theorem, the class number of such extensions should behave similarly to those of imaginary quadratic fields, in particular given fixed $$n$$ only finitely many solutions $$d$$ exists for $$h(d) = n$$. In fact, conjectures on the finiteness of subfamilies with class number one of R-D type was conjectured by Yokoi ($$a = 1$$, $$k = 4$$), Chowla ($$n = 2m$$, $$ka = 1$$), Mollin ($$a = 1$$, $$k = -4$$). These conjectures have been proven by Biro, Lapkova, and Byeon et.al after the breakthrough by Biro in 2003.

However, my question is whether it is known that there are infinitely many such square free $$d$$ of Richaud-Degert type.

Also, I would like to hear any opinions about the impact of these results on the general class number problem of real quadratic fields. Any speculations as to where these results may lead us in relation to the general class number problem of real quadratic fields are welcome. Are there any recent (past 5 years) advancements in this regard?

Watkins has recently (2019) improved Goldfeld's lower bound for $$L_\chi(1)$$ by a constant, by using elliptic curves of rank 5 and computing $$L'''_E(1)$$ to 1000 digits.

This allows the explicit solution for class number problems up to (say) 5 for various families with small fundamental unit.

Edit: he has a newer version of this preprint Class Number Problems For Real Quadratic Fields With Small Fundamental Unit

• Also, something is wrong with your definition of R-D fields, it should be $-n<k\le n$, not $-n<d\le n$. Furthermore, the fundamental unit is of size $\sqrt d$, not $d$. – user174996 Feb 27 at 23:03
• Thank you for pointing it out, edited the post to correct it. – Melanka Feb 27 at 23:07

"However, my question is whether it is known that there are infinitely many such square free $$d$$ of Richaud-Degert type."

If I understand this correctly, it suffices to show that $$4m^2+1$$ (Chowla's family) is infinitely often squarefree.

Then this (infinitely many squarefree values of a quadratic polynomial) follows from work of Nagell (1922). http://dx.doi.org/10.1007/BF02940590

See references from Rudnick's course notes: http://www.math.tau.ac.il/~rudnick/courses/sieves2015/squarefrees.pdf