All Questions

Consider an integer of the form $$N = 1 + \sum_{i=1}^r d_i^2$$ where $d_i \in \mathbb{N}_{\ge 3}$ and $d_i^2$ divides $N$. Question: Must $r$ be greater than or equal to $9$? Checking (with SageMath): ...

Recently I've been studying the problem of integer representation as sum of three squares. Most of the articles that I've found study the function $r_m(n)$ which counts the number of representations ...

Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...

I need to compute efficiently the sum $$ \sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor. $$ We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it ...

Hello, The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics) For all integers $n\geq 2$ denote by $\mathcal{P}...

Sum of 4 squares decomposition is the well-known result. I'm interested only in negative/non-negative separation with focus on efficiency and large numbers. I'm looking for alternatives or extensions ...