TITLE: Euclidean real quadratic fields
QUESTION [10 upvotes]: It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic fields. 
I want to know if there are any approaches in showing that there are infinitely many Euclidean real quadratic fields by explicitly constructing the Euclidean functions. Can we hope for a single (or finitely many) Euclidean function(s) that will do the job or we need infinitely many different Euclidean functions? Can these Euclidean functions be obtained by modifying the norm function?    
Since very few non-norm Euclidean quadratic fields are known, can we at all hope for showing the infinitude of real UFD quadratic fields by finding infinitely many Euclidean quadratic fields?

REPLY [5 votes]: The problem of finding explicit Euclidean functions is even more complicated.
Usually the euclidean property is proved indirectly, using sieve-theoretic methods (in particular, Wilson's large sieve over number fields and the Gupta-Murty lemma). For this kind of argument see Harper's work on $\mathbb{Q}(\sqrt{14})$ and Harper-Murty.
I think that the only known non-trivial example (that is, for fields which are not norm-Euclidean) of a real quadratic field with an Euclidean function  is $\mathbb{Q}(\sqrt{69})$:

D. A. Clark, "A quadratic field which is euclidean but not norm-euclidean" (1994)

"Since very few non-norm Euclidean quadratic fields are known" I don't think this is true, depending on the meaning of "few" here. Malcolm Harper showed in his thesis that all real quadratic fields with class number $1$ and discriminant $\leq 500$ are Euclidean, but I believe this was never published.