View source for Hilbert number

{{Short description|A positive integer of the form (4n + 1)}}
{{No footnotes|date=December 2024}}
{{about|the sequence 1, 5, 9, 13, ...|<math>2^{\sqrt{2}}</math>|Gelfond–Schneider constant}}
In [[number theory]], a branch of [[mathematics]], a '''Hilbert number''' is a [[positive integer]] of the form {{math|4''n'' + 1}} ({{harvtxt|Flannery|Flannery|2000|p=35}}). The Hilbert numbers were named after [[David Hilbert]].
The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... {{OEIS|id=A016813}})

==Properties==
*The Hilbert number sequence is the [[arithmetic sequence]] with <math>a_1=1,d=4</math>, meaning the Hilbert numbers follow the [[recurrence relation]] <math>a_n=a_{n-1}+4</math>.
*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, ... {{OEIS|id=A057948}}.
A Hilbert prime is not necessarily a [[prime number]]; for example, 21 is a [[composite number]] since {{math|1=21 = 3 &sdot; 7}}.  However, 21 {{em|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 [[modular arithmetic|modulo]] 4 that a Hilbert prime is either a prime number of the form {{math|4''n'' + 1}} (called a [[Pythagorean prime]]), or a [[semiprime]] of the form {{math|(4''a'' + 3) &sdot; (4''b'' + 3)}}.

==References==
*{{citation|first1=S.|last1=Flannery|first2=D.|last2=Flannery|title=In Code: A Mathematical Journey|publisher=Profile Books|year=2000}}

==External links==
* {{mathworld|urlname=HilbertNumber|title=Hilbert Number}}
* {{OEIS el|A057949|Numbers with more than one factorization into Hilbert primes}}

{{Classes of natural numbers}}

[[Category:Eponymous numbers in mathematics]]
[[Category:Integer sequences]]


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