Hilbert number

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

The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... (sequence A016813 in the 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 begins

5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948 in the OEIS).

A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 7. However, 21 is a Hilbert prime since neither 3 nor 7 (the only factors of 21 other than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form 4n + 1 (called a Pythagorean prime), or a semiprime of the form (4a + 3) (4b + 3).

References

  • Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.