It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.

Kasteleyn's formula reveals $\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\right)$ is an integer because it enumerates the domino tilings of a $2n$-by-$2n$ square.

These results prompt me to ask for more. First, let's introduce the $r$-product $$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left( 4\cos^2\left(\frac{\pi\ell_1}{2n+1}\right)+\cdots+4\cos^2\left(\frac{\pi\ell_r}{2n+1}\right)\right).$$

Questions.This is based on experimental assesment.(a) Is $K_r(n)$ always an integer?

(b) Is there perhaps a higher-dimensional combinatorial interpretation of Kasteleyn for $K_r(n)$?

(c) Why do $K_r(n)$ feature "small primes" with high-power factorizations? For example, $K_3(2)=3^2(19)^3,\,\, K_3(3)=3^35^6(83)^3(97)^3\,\, K_3(4)=2^63^{34}(17)^6(19)^6(37)^6\,\,$ and $$K_3(5)=3^5(43)^6(1409)^3(2267)^3(2707)^3(3719)^6(3389)^6.$$

I'm not aware of such a generalization, but any reference would be appreciated. Thanks.

**UDPATE** The comments have answered (a). Is there a more direct (elementary) proof? Any suggestions for parts (b) and (c)?