Centered cube number

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 trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as^[2]

${\displaystyle {\binom {(n+1)^{2}+1}{2}}-{\binom {n^{2}+1}{2}}=(n^{2}+1)+(n^{2}+2)+\cdots +(n+1)^{2}.}$

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.

• Cube number