In
number theory, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Hilbert number is a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
of the form (). The Hilbert numbers were named after
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
.
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
: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 . 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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
4 that a Hilbert prime is either a prime number of the form (called a
Pythagorean prime), or a
semiprime of the form .
References
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External links
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*
{{Classes of natural numbers
Integer sequences