# .

In number theory, a Hilbert number is defined as a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert.

The integer sequence of Hilbert numbers is 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, … (sequence A016813 in OEIS). A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes is 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (OEIS A057948). Note that Hilbert primes do not have to be prime numbers; for example, 21 is a composite Hilbert prime. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of form 4n + 1 (called a Pythagorean prime), or a semiprime of form (4a + 3) × (4b + 3).

References

Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books