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?

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