number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Hilbert number is a

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

of the form (). The Hilbert numbers were named after

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

. 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 In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

a_n=a_+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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

; for example, 21 is a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

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 and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

4 that a Hilbert prime is either a prime number of the form (called a Pythagorean prime), or a semiprime of the form .

References

External links

* * Eponymous numbers in mathematics Integer sequences {{Num-stub