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Setting the finite sum of squares equal to a square number yields only one non-trivial solution, when $n= 24$; the sum becomes $4900$, which is $70^2$ $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ https://youtu.be/vzjbRhYjELo?t=4m21s

The infinite sum of reciprocal squares is equivalent to evaluating the Riemann Zeta Function at $s=2$. $Zeta(2) = \frac{\pi^2} 6$

https://www.youtube.com/watch?v=d-o3eB9sfls

If you replace $\pi$ with $\tau (\tau = 2*\pi = 6.2931...)$ we can see that $Zeta(2)$ equals $\frac {(\tau^2)}{24}$

$24$ appears in both summations of squares. One way is a finite sum starting at the apex of a square pyramid toward an infinitely far away base, and the other problem involves starting at the base and going up towards an infinitely far away apex.

What is happening with the number $24$ and this "inversion" of the summations of these square pyramids?!

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    $\begingroup$ I dunno. Either something humble like 24 just happens to be product of a few small primes and occurs relatively often by a harmless accident..... $\endgroup$ – Jyrki Lahtonen Mar 19 '18 at 20:36
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    $\begingroup$ ... OR YOU PETTY PEDDLER! TURN IT UP TO 11. OF COURSE IT'S 24. 24 is the answer to everything. Douglas Adams got it all wrong. Look up Monstrous Moonshine, and the role the Leech Lattice played (a super tight packing that makes algebraists and coding theorists drool and only exists in dimension 24) $\endgroup$ – Jyrki Lahtonen Mar 19 '18 at 20:40
  • $\begingroup$ Finite Sum (n=24): 1²+2²+3²+...+24² = 70² Infinite Sum (n→inf): 1+(1/2)²+(1/3)²+... = pi²/6 = tau²/24 Is there anyway to infer the solution of one problem to the other one using the idea of the perspective change... where we start from the square pyramid's base or apex? If one can imagine an infinite pyramid starting from the top-down, we can solve the diophantine equation for (24,70). If we imagine starting at the base and going up smaller and smaller levels to an apex at infinity, we derive the well known solution for Zeta(2). What is it about the perspective flip from apex to base? $\endgroup$ – Jordan Blevins Mar 19 '18 at 23:19
  • $\begingroup$ @JyrkiLahtonen Were you referencing "This is Spinal Tap" when writing "turn it up to 11?" $\endgroup$ – Mark Viola Mar 20 '18 at 2:34
  • $\begingroup$ @MarkViola Yes. I'm not good at all with pop-culture references. The thinking was a bit like: if we go into numerology, and believe in the holy 24, then let's go all the way, and go into as deep math as we can. In the case of 24 only one direction occurred to me .... $\endgroup$ – Jyrki Lahtonen Mar 20 '18 at 5:12

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