Editing Centered nonagonal number

{{Short description|Centered figurate number that represents a nonagon with a dot in the center}}
[[File:Centered nonagonal number.svg|280px|right]]

A '''centered nonagonal number,''' (or '''centered enneagonal number'''), is a [[centered number|centered]] [[figurate number]] that represents a [[nonagon]] with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered [[nonagonal number]] for ''n'' layers is given by the formula<ref name="oeis">{{Cite OEIS|A060544|Centered 9-gonal (also known as nonagonal or enneagonal) numbers}}</ref>

:<math>Nc(n) = \frac{(3n-2)(3n-1)}{2}.</math>

Multiplying the (''n'' - 1)th [[triangular number]] by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.<ref name="oeis"/>

Thus, the first few centered nonagonal numbers are<ref name="oeis"/>
:[[1 (number)|1]], [[10 (number)|10]], [[28 (number)|28]], [[55 (number)|55]], [[91 (number)|91]], [[136 (number)|136]], [[190 (number)|190]], [[253 (number)|253]], [[325 (number)|325]], 406, [[496 (number)|496]], 595, 703, 820, 946.

The list above includes the [[perfect number]]s 28 and 496.
All [[even number|even]] perfect numbers are triangular numbers whose index is an odd [[Mersenne prime]].<ref>{{citation|title=Pell and Pell–Lucas Numbers with Applications|first=Thomas|last=Koshy|publisher=Springer |isbn=9781461484899|year=2014|page=90|url=https://books.google.com/books?id=j_9MBQAAQBAJ&pg=PA90}}.</ref> Since every Mersenne prime greater than 3 is congruent to 1&nbsp;[[Modular arithmetic|modulo]]&nbsp;3, it follows that every even perfect number greater than&nbsp;6 is a centered nonagonal number.

In 1850, [[Sir Frederick Pollock, 1st Baronet|Sir Frederick Pollock]] conjectured that every natural number is the sum of at most eleven centered nonagonal numbers.<ref>{{citation|authorlink=L. E. Dickson|last=Dickson|first=L. E.|series=[[History of the Theory of Numbers]]|volume=2|title=Diophantine Analysis|location=New York|publisher=Dover|year=2005|pages=22–23|isbn=9780821819357 |url=https://books.google.com/books?id=eNjKEBLt_tQC&pg=PA22}}.</ref> Pollock's conjecture was confirmed as true in 2023.<ref>{{Cite journal |last=Kureš |first=Miroslav |date=2023-10-27 |title=A Proof of Pollock's Conjecture on Centered Nonagonal Numbers |url=https://link.springer.com/10.1007/s00283-023-10307-0 |journal=The Mathematical Intelligencer |language=en |doi=10.1007/s00283-023-10307-0 |issn=0343-6993}}</ref>

==Congruence Relations==
*All centered nonagonal numbers are congruent to 1 mod 3.
**Therefore the sum of any 3 centered nonagonal numbers and the difference of any two centered nonagonal numbers are divisible by 3.

==See also==
*[[Nonagonal number]]

==References==
{{reflist}}

{{Figurate numbers}}
{{Classes of natural numbers}}

{{DEFAULTSORT:Centered Nonagonal Number}}
[[Category:Figurate numbers]]