Centered nonagonal number

A centered nonagonal number, (or centered enneagonal number), is a 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 ^[1]

${\displaystyle Nc(n)={\frac {(3n-2)(3n-1)}{2}}.}$

Multiplying the (n − 1)th triangular number by 9 and then adding 1 yields the nth 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. ^[1]

Thus, the first few centered nonagonal numbers are ^[1]

1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946.

The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. ^[2] Since every Mersenne prime greater than 3 is congruent to 1  modulo  3, it follows that every even perfect number greater than 6 is a centered nonagonal number.

In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers. ^[3] Pollock's conjecture was confirmed as true in 2023. ^[4]

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

1. Sloane, N. J. A. (ed.). "SequenceA060544(Centered 9-gonal (also known as nonagonal or enneagonal) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Koshy, Thomas (2014), Pell and Pell–Lucas Numbers with Applications, Springer, p. 90, ISBN   9781461484899 .
3. Dickson, L. E. (2005), Diophantine Analysis, History of the Theory of Numbers, vol. 2, New York: Dover, pp. 22–23, ISBN   9780821819357 .
4. Kureš, Miroslav (2023-10-27). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN   0343-6993.