Hilbert number

From Wikipedia, the free encyclopedia

In mathematics, Hilbert number, named after David Hilbert, has different meanings.

In analysis and number theory, the Hilbert number (also called the Gelfond-Schneider constant), is the mathematical constant $2^{\sqrt{2}}$ (Courant & Robbins (1996), p. 107)). This has been proved to be a transcendental number.

In number theory, a Hilbert number is defined as a positive integer of the form 4n + 1 (Flannery & Flannery (2000), p. 35)). 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, ... (A057948). Note that Hilbert primes do not have to be prime numbers. 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).

[edit] References

Courant, R. & Robbins, H. (1996), What Is Mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press
Flannery, S. & Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books