Editing Centered polygonal number (section)

== Formulas ==

As can be seen in the above diagrams, the ''n''th centered ''k''-gonal number can be obtained by placing ''k'' copies of the (''n''&minus;1)th triangular number around a central point; therefore, the ''n''th centered ''k''-gonal number is equal to

:<math>C_{k,n} =\frac{kn}{2}(n-1)+1.</math>

The difference of the ''n''-th and the (''n''+1)-th consecutive centered ''k''-gonal numbers is ''k''(2''n''+1).

The ''n''-th centered ''k''-gonal number is equal to the ''n''-th regular ''k''-gonal number plus (''n''-1)<sup>2</sup>.

Just as is the case with regular polygonal numbers, the first centered ''k''-gonal number is 1. Thus, for any ''k'', 1 is both ''k''-gonal and centered ''k''-gonal. The next number to be both ''k''-gonal and centered ''k''-gonal can be found using the formula:

:<math>\frac{k^2}{2}(k-1)+1</math>

which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.

Whereas a [[prime number]] ''p'' cannot be a [[polygonal number]] (except the trivial case, i.e. each ''p'' is the second ''p''-gonal number), many centered polygonal numbers are primes. In fact, if ''k'' ≥ 3, ''k'' ≠ 8, ''k'' ≠ 9, then there are infinitely many centered ''k''-gonal numbers which are primes (assuming the [[Bunyakovsky conjecture]]). Since all [[centered octagonal number]]s are also [[square number]]s, and all [[centered nonagonal number]]s are also [[triangular number]]s (and not equal to 3), thus both of them cannot be prime numbers.