Centered cube number

Centered cube number

Polyhedral numbers
Formula	${\displaystyle n^{3}+(n+1)^{3}}$
First terms	559
OEIS index	A005898 Centered cube

A centered cube number is a

The first few centered cube numbers are

9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... (sequence A005898 in the OEIS

).

The centered cube number for a pattern with n concentric layers around the central point is given by the formula[1]

${\displaystyle n^{3}+(n+1)^{3}=(2n+1)\left(n^{2}+n+1\right).}$

The same number can also be expressed as a

Because of the factorization (2n + 1)(n² + n + 1), it is impossible for a centered cube number to be a prime number.^[3] The only centered cube numbers which are also the square numbers are 1 and 9,^[4][5] which can be shown by solving x² = y³ + 3y , the only integer solutions being (x,y) from {(0,0), (1,2), (3,6), (12,42)}, By substituting a=(x-1)/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1. This gives only (a,b) from {(-1/2,0), (0,1), (1,3), (11/2,21)} where a,b are half-integers.

See also

• Cube number

References