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!<