Hilbert number

In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form (). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... )

Properties

• The Hilbert number sequence is the arithmetic sequence with , meaning the Hilbert numbers follow the recurrence relation .
• The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number.

Hilbert primes

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

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

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