Hilbert number

From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the sequence 1, 5, 9, 13, ... For 2^{\sqrt{2}}, see Gelfond–Schneider constant.

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, ... (OEISA057948). 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[edit]

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

External links[edit]