It is my understanding that a Hilbert number is any number of the form $4n + 1$.

According to


it seems that these numbers were named in honor of Hilbert almost twenty years ago.

My question is: Does anyone know why; or what connection they may have had to David Hilbert who was not a number theorist; and so, I am imagining, did not study them.

Or did he?

Does anyone know for sure?


The context, I am told, is that Hilbert wanted to illustrate that the uniqueness of prime factorization is not a completely obvious claim that needs no proof.

To this end, we introduce Hilbert numbers (which indeed have a rather silly definition out of context) and Hilbert primes (which are Hilbert numbers $n$ whose only Hilbert number divisors are $1$ and $n$ itself). Hilbert primes are not Hilbert numbers which are prime in the ordinary sense: for example, $9$ is a Hilbert prime because its only nontrivial factor, $3$, is not a Hilbert number.

Every Hilbert number has a "prime factorization" into Hilbert primes. But that factorization is not unique; for example, $441$ has two prime factorizations $$441 = 21 \cdot 21 = 9 \cdot 49.$$

(I'm not positive that Hilbert himself used this as an example; I can't find a source for this.)


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