So after I learned this two topic: quantization and sampling, I'm learning the way to look at both of them and try to optimize the split of a given amount of bit B to N and k, where N is the amount of samples and k is the size of the finite set of numbers can represent the values (quantization). ...

P. Erdos and J. L. Selfridge proved in the paper THE PRODUCT OF CONSECUTIVE INTEGERS IS NEVER A POWER (click here), that the equation $(n + 1) \cdots(n + k)=x^l \cdots (1)$ has no solution in integers with $k > 2, l > 2, n > 0$. There is a lemma $1$, I have $2$ issues (for better searchability f...

P. Erdos and J. L. Selfridge proved in the paper THE PRODUCT OF CONSECUTIVE INTEGERS IS NEVER A POWER (click here), that the equation $(n + 1) \cdots(n + k)=x^l \cdots (1)$ has no solution in integers with $k > 2, l > 2, n > 0$. There is a lemma $2$ on page 294, 295 - LEMMA 2. By deleting a sui...

Given,$$D(G) ≤ \exp(G)(1 + \log(|G|/\exp(G)))$$, where $D(G)$ is Davenport's constant, for any finite Abelian group $G$ . Let the cyclic group of the order $w$ is denoted by $C_w$. If $$ g_1, \dots, g_k \in C_w^t, k ≥ tw \log(w), G = C_w^t,$$ and $$\exp(G) = w$$, then how the below inequal...

Let, $\Pi_{n,m}= n(n+1) \cdots (n+m-1)$ and $P (m)$ denotes greatest prime divisor of a positive integer $m$. For fixed $m$ and $t$, we will in fact consider the problem of classifying those positive integers $n$ for which $(2.1) \; \; P(\Pi_{n,m)} ≤ p_t$ where $p_t$ is the $t^{th}$ prime. we wil...