Hilbert number

In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form $4n + 1$. 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 $$a_1=1,d=4$$, 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.

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


 * 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ....

A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since $21 = 3 &sdot; 7$. 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 $4n + 1$ (called a Pythagorean prime), or a semiprime of the form $(4a + 3) &sdot; (4b + 3)$.