Define the congruence "modulo m" on exponential Taylor series following the previous post (A conjecture about the degrees of special polynomials)
It has been conjectured, that if we define the sequence of polynomials $Q_{n,k}(x)$: $$ Q_{n,k}(x):= \left(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...+\frac{x^{p_n-1}}{p_n-1}\right)^k \mod p_n! $$ then $\forall n\in \mathbb{N}, 2<k<p_{n-1}:$ $$ \boxed{\deg Q_{n,k}(x)=k\cdot p_{n-1}} $$ (where $p_n$ is the n-th prime number and so $p_{n-1}$ is the previous one)
It is also conjectured to be true for $k=2$ and all $n\neq 1,2,5~(p_n\neq 2,3,11)$.
I still have no idea, why the degrees of polynomials of this kind have such "number theoretical" properties and would appreciate, if you could give a proof of this fact, maybe share philosophical speculations about the nature of this structure or post some references. Thank you.