I'm looking for an efficient way to count how many non-negative integer solutions (X,Y) satisfy the formula: X^2 + Y^2 = 2*N^2 for a given positive integer N I know of Pell's eq …

Every prime $p = 4k + 1$ can be uniquely expressed as sum of two squares, but for which integers $x$ is $x^2 + y^2 =$ some prime $p$? Stated differently, does the square of every …

Hi, I've been studying some proofs of The four squares theorem. Some of them are pretty clear. However, I came across a statement that The four squares theorem can be easily derive …

Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i, …

While tinkering with numbers, I found $n=f(x)$ where $n = \sigma(\sigma(n)-n)$ and $x \neq p$ where $p$ is a Mersenne prime exponent, but I need help input in regard to improving a …

Hi I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some …

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 $n\geq 2$ d …

Might the following be true: Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$ …

What happens if we follow the construction of the Leech lattice but replace the relation $\displaystyle \sum_{n=1}^{24} n^2 = 70^2$ with the smallest perfect squared square? Exp …

Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squar …

The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed to …

Define the quadratic form $$Q(z_1,z_2,z_3,z_4) = 13 + \sum_{i=1}^4 (10+i)z_i +5 \sum_{1 \le i \le j \le 4} z_iz_j.$$ Then, $r_Q(n) := \left|{(z_1,z_2,z_3,z_4) \in \mathbb{Z}^4 : …

It seems that there are three basic ways to prove an inequality eg $x>0$. Show that x is a sum of squares. Use an entropy argument. (Entropy always increases) Convexity. Are th …

It's known that the number of representations of an integer $k$ by sum of two squares is $$ 4\;\sum_{d|k}\left(\frac{-4}{d}\right) $$ or $$ 4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\fr …

Is there a general method of determining the line of best fit (using the principle of least squares or any other principle) for any given set of data points$? If there is no genera …