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): ...

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 ...

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 \...

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 ...

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}...

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 ...