A centered nonagonal number, (or centered enneagonal number), is a centered

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...

that represents a

nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix Hybrid word, hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in Fre ...

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 :

Nc(n) = \frac.

Multiplying the (''n'' - 1)th

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

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. Thus, the first few centered nonagonal numbers are : 1, 10, 28, 55, 91,

136 136 may refer to: *136 (number) *AD 136 *136 BC *136 (MBTA bus), a Massachusetts Bay Transportation Authority bus route *136 Austria 136 Austria is a main-belt asteroid that was found by the prolific asteroid discoverer Johann Palisa on 18 Ma ...

, 190, 253, 325, 406, 496, 595, 703, 820, 946. The list above includes the

perfect number 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 perfec ...

s 28 and 496. All even perfect numbers are triangular numbers whose index is an odd

Mersenne prime 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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

. Since every Mersenne prime greater than 3 is congruent to 1

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 ...

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. Pollock's conjecture was confirmed as true in 2023.

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.

References

{{DEFAULTSORT:Centered Nonagonal Number Figurate numbers

Congruence Relations

See also

References