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Let be a number field, i.e. a finite field extension to . We recall that the * ring of integers* in

*k*,

*denoted , is the ring*

For , the ring of integers is just the integers , in which case we recall the Fundamental Theorem of Arithmetic: that every integer may be written as a finite product

in which the are prime and uniquely determined (up to permutation). Domains for which this holds are known in general as ** unique factorization domains **(UFDs). For — with square-free — the ring of integers will in general

*not*be a UFD. In fact, for , the integers have unique factorization only in the 9 cases

Far less is known in the case (in which case *k* is known as a * real quadratic field*), although an unproven conjecture dating back to Gauss suggests that there should be infinitely many real quadratic fields. More recently, some heuristics stemming from Cohen suggests that the ring of integers in should be a UFD with probability as on the square-free integers.

Here, we’ll focus on a more tractable variant of this problem:

**Question:** What can be said about the number of distinct real quadratic fields with for which is ** not** a UFD?

For a weak answer to the question above, we devote the rest of this article to the establishment of the following bound:

**Theorem: **As , we have

in which the implied constant is made effective (e.g. greater than ).