Hilbert number

In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form $4 n + 1$ (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))

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, ... (sequence A057948 in the OEIS).

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

References

Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books

External links

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