A clear piece on the study of class numbers (a bit of math) via *Quanta*

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the German mathematician Ernst Kummer developed a way to fix the loss of prime factorization with what he called “ideal numbers.” They’re not numbers in any conventional sense. Rather, they’re sprawling constructions in set theory that perform a number-like function.

For example, the simplest ideal is the infinite set of all multiples of a given integer — 5, 10, 15, 20 and so on. Ideals can be added into an already expanded number ring to restore unique factorization. They allow mathematicians to reconcile competing prime factorizations into a single set of prime factors.

Ideals can be categorized into various classes. The number of different classes of ideals you need to add to a number ring in order to restore unique factorization is the ring’s “class number.”

The study of class numbers goes at least as far back as Carl Friedrich Gauss in the early 19th century. In a sign of how hard it’s been to make progress in this area, many of his results are still state of the art. Among his contributions, Gauss conjectured that there are infinitely many positive square roots that can be adjoined to the whole numbers without losing unique factorization — a proof of which remains the most sought-after result in class numbers (and is rumored to have frustrated Kurt Gödel enough to make him give up number theory for logic). Gauss also conjectured that there are only nine negative square roots that preserve prime factorization. √-163 is the very last one.

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