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# Hilbert numbers

The Hilbert numbers or S-numbers are numbers of the form ${\displaystyle 4n+1}$.

A016813 4n+1.

{1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, ...}

Though this set is closed under multiplication (the product of Hilbert numbers is another Hilbert number), it does not form a unique factorization domain[1] (and in any case it does not form a ring as it lacks a zero).[2]

## Hilbert primes

The Hilbert primes or S-primes are Hilbert numbers not divisible by any smaller Hilbert number except 1.

A057948 S-primes: let S = {1, 5, 9, ..., 4i+1, ...}; then an S-prime is in S but is not divisible by any members of S except itself and 1.

{5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 229, ...}

## Hilbert composites

The Hilbert composites or S-composites are then 25, 45, 65, 81, 85, 105, 117, 125, ... (A054520 lists the S-nonprimes and therefore starts 1, 25, 45, etc.)

A054520 Let S = {1, 5, 9, 13, ..., 4n+1, ...} and call p in S an S-prime if p > 1 and the only divisors of p in S are 1 and p; sequence gives elements of S that are not S-primes.

{1, 25, 45, 65, 81, 85, 105, 117, 125, 145, 153, 165, 169, 185, 189, 205, 221, 225, 245, 261, 265, 273, 285, 289, 297, 305, 325, 333, 345, 357, 365, 369, 377, 385, 405, 425, 429, ...}

Numbers that are Hilbert primes but composites in ${\displaystyle \scriptstyle \mathbb {Z} \,}$ are the product of two primes of the form 4n+3 (we can verify that ${\displaystyle \scriptstyle (4m+3)(4n+3)\,=\,16mn+12m+12n+9\,=\,4s+1\,}$; therefore neither prime is a Hilbert number and their product is not divisible by any smaller Hilbert number. These Hilbert primes are listed in A107978.

Keeping in mind that 9, 21 and 49 are Hilbert primes, we see that 441 does not have a unique factorization into Hilbert primes, since ${\displaystyle \scriptstyle 9\times 49=21^{2}\,=\,441\,}$. See A057949 for more examples.

## Notes

1. P. Giblin, Primes and Programming: An Introduction to Number Theory with Computing, Cambridge University Press (1993) p. 30.
2. Weisstein, Eric W., Ring, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Ring.html]. (Conditions 3 and 4 are identified as “always required” and both involve 0.)