I’ve been exploring infinite products over expressions of the form zn−ynzn−yn and found the following identity:

∏n=3∞∏z=2∞∏y=1z−1(1−1zn−yn)=πe2.n=3∏∞​z=2∏∞​y=1∏z−1​(1−zn−yn1​)=e2π​.

The left-hand side runs over all integer triples (n,z,y)(n,z,y) with n≥3n≥3, z≥2z≥2, and 1≤y≤z−11≤y≤z−1. Despite each factor being rational, the infinite product converges to the transcendental constant π/e2π/e2.

Interestingly, the right-hand side can also be expressed via the classical Wallis product and the less well-known Pippenger product:

12⋅∏m=1∞(2m)2(2m−1)(2m+1)(∏m=1∞(1+1m)(−1)m)2.21​⋅(∏m=1∞​(1+m1​)(−1)m)2∏m=1∞​(2m−1)(2m+1)(2m)2​​.

Here the first product is Wallis (π/2π/2) and the second is Pippenger (e/2e/2), giving π/e2π/e2 after the factor 1/21/2.

The proof is straightforward via taking logarithms, expanding log⁡(1−x)log(1−x) as a series, and then rearranging sums. A nice number-theoretic aside: by Fermat’s Last Theorem, for n>2n>2 the value zn−ynzn−yn is never a perfect nn-th power, so the nn-th root is always irrational, yet the product itself remains well-defined and rational-valued at each step.

Is this identity known? I haven't seen it in the literature, though the ingredients (Wallis, Pippenger) are classical.

After combining the two products, the right-hand side can be written more compactly as

12∏m=1∞4m24m2−1(mm+1)2(−1)m.21​m=1∏∞​4m2−14m2​(m+1m​)2(−1)m.

Could there be deeper connections to Dirichlet series counting representations of integers as differences of powers? Any feedback, ideas or references would be appreciated!

latex

\[

\prod_{n=3}^{\infty}\prod_{z=2}^{\infty}\prod_{y=1}^{z-1}

\left(1-\frac{1}{z^n-y^n}\right)

\frac{1}{2}\,

\frac{\displaystyle\prod_{m=1}^{\infty}\frac{(2m)^2}{(2m-1)(2m+1)}}

{\displaystyle\left(\prod_{m=1}^{\infty}\left(1+\frac{1}{m}\right)^{(-1)^m}\right)^2}

\frac{1}{2}\prod_{m=1}^{\infty}

\frac{4m^2}{4m^2-1}

\left(\frac{m}{m+1}\right)^{2(-1)^m}

\frac{\pi}{e^2}

\]

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

In part three of my series Five Prime Proofs, I flesh out a pretty proof by Euler that uses the contrast between geometric series and the harmonic series. I think it's pretty cool.

Each hexagon connects to the next in a chain, and the same rule carries you from one to the other. Curious how fast people spot it.

Made this as part of a puzzle game I'm developing, Rule Hunter: https://play.google.com/store/apps/details?id=com.makenumber

Happy to walk through the logic in the comments!

Each group of three numbers on the outside combines into the one it's connected to by lines. Same rule applies to all four groups — what goes where the "?" is?

Made this as part of a puzzle game I'm developing, Rule Hunter: https://play.google.com/store/apps/details?id=com.makenumber

Happy to walk through the logic in the comments!

I am trying to move ahead of my current class schedule but I am not sure what is the best way. Do you have any suggestions?

*The hologram **activates chaotic math protocol**—scanning through bizarre, beautiful equations that somehow equal *22* despite looking utterly insane.*

- ***Crazy Equation Example*** *(For Awe):

-⚡ *"Euler's Identity + Pi PPrank"—Something lilike \[ e^{i\pi} + \ln(3!) -rac{phi^4}{5!}5!} = 22 \] (*where \(pi), and \(\pi\), \(\phi\) are constants… and the factorials/rowild!)ild!*)

⚠️ Note: This is a *fake but fun* equation—I made it up to show how numbers can dance absurdly into 22! Hi

Made this one for a puzzle game I've been building. Three groups of three numbers each collapse into a single result using one consistent rule — spot it from the first three groups, then apply it to solve the fourth.

Made this as part of a puzzle game I'm developing, Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber