Cornered Number

Algebra Level 2

Let $P_n$ denote the $n^{th}$ square pyramidal number. $P_n$ is a square pyramidal number that is simultaneously a square number where $n$ is greater than $1!$ . $\phi(\phi(P_n)) = k$ where $\phi$ is Euler's totient function.

Find the decimal value of $|k\sqrt{P_n}\div n - n^3|$ where $| |$ is the absolute value function.

The answer is 12704.

1 solution

Mohammad Farhat
Oct 15, 2018

This is VERY closely related to the cannonball problem: Find a way to stack a square of cannonballs laid out on the ground into a square pyramid.

This corresponds to solving the Diophantine equation:

$\sum_{i=1}^k i^2 = \frac{1}{6} k(1+k)(1+2k) = N^2$ for some pyramid height $k$

The only solutions of $(k,N)$ are :

$(1,1)$

$(24,70)$

I leave the rest to you because I already gave you the variables and you can just plug them in.

Cornered Number

The answer is 12704.

1 solution

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