Hilbert number

Template:Short description Template:No footnotes Template:About In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form Template:Math (Template:Harvtxt). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... Template:OEIS)

Properties

The Hilbert number sequence is the arithmetic sequence with $a_{1} = 1, d = 4$ , meaning the Hilbert numbers follow the recurrence relation $a_{n} = a_{n - 1} + 4$ .
The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number.

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, ... Template:OEIS.

A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since Template:Math. However, 21 Template:Em 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 Template:Math (called a Pythagorean prime), or a semiprime of the form Template:Math.