Centered nonagonal number

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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]

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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

See also

Related Research Articles

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.

<span class="mw-page-title-main">Prime number</span> Number divisible only by 1 or itself

A prime number is a natural number greater than 1 that is not a 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 because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

<span class="mw-page-title-main">Perfect number</span> Integer equal to the sum of its proper divisors

In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.

28 (twenty-eight) is the natural number following 27 and preceding 29.

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

91 (ninety-one) is the natural number following 90 and preceding 92.

500 is the natural number following 499 and preceding 501.

2000 is a natural number following 1999 and preceding 2001.

<span class="mw-page-title-main">Hexagonal number</span> Type of figurate number

A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.

<span class="mw-page-title-main">Pentagonal number</span> Figurate number

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

3000 is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English.

4000 is the natural number following 3999 and preceding 4001. It is a decagonal number.

5000 is the natural number following 4999 and preceding 5001. Five thousand is, at the same time, the largest isogrammic numeral, and the smallest number that contains every one of the five vowels in the English language.

6000 is the natural number following 5999 and preceding 6001.

8000 is the natural number following 7999 and preceding 8001.

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered k-gonal number contains k more dots than the previous layer.

A nonagonal number, or an enneagonal number, is a figurate number that extends the concept of triangular and square numbers to the nonagon. However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the nth nonagonal number counts the dots in a pattern of n nested nonagons, all sharing a common corner, where the ith nonagon in the pattern has sides made of i dots spaced one unit apart from each other. The nonagonal number for n is given by the formula:

204 is the natural number following 203 and preceding 205.

253 is the natural number following 252 and preceding 254.

References

  1. 1 2 3 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 (October 27, 2023). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN   0343-6993.
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