Determine all the representations of the integer $2130797 = 17^2 \times 73 \times 101 $ as a sum of two squares. attempt: Suppose $2130797$ is of the form $n = 2^kp_1^{a_1}....p_r^{a_r}q_1^{b_1}...q_s^{b_s}$ . Where $p_1,...,p_r$ are distinct primes congruent to $1 $ modulo $4$ and $q_1,....,q_s...

Let $\mathbb{D} = \{s \in \mathbb{C} : |s| < 1\}$. Let $f : \mathbb{D} \rightarrow \mathbb{C}$ where $$ f(s) = \sum_{n=0}^\infty s^{n^2} $$ $f$ is analytic on $\mathbb{D}$. This is what it looks like: $\partial \mathbb{D}$ is a natural boundary, so ordinary analytic continuation cannot extend $f...

The following code demonstrates that for all $n \geq 4$, there exists an interval $I = [p, q]$ such that $|I|$ is maximal with $I \subset Y$ and $p, q$ are both prime numbers. Therefore, the maximal size $|I|$ is (probably) bounded by the maximal prime gap in the interval $[0, m)$ where $m = p_...

So I'm working through the problems in this book. While I'm not at school, I can totally see how this might be someones homework, so if I have made any egregious errors in anything outside of the specific question, be a sport and don't blurt it out. I should stress, I haven't taken a calculus cou...

I am reading the book "Analytic S-matrix" by Eden,Landshoff, Olive and Polkinghorne. In Sec.2, they discuss the different kinds of singularities of a function $f(z)$ defined as an integral of an analytic function $g(z,\omega)$ in the complex $\omega-$plane along some closed contour $\mathcal{C}$,...

Consider the union $U_N = \bigcup\limits_{i=1}^N (p_i\Bbb{Z} \pm 1)$, where $p_i$ is the $i$th prime number. Define a measurement $\mu(N)$ on $U_N$ namely the maximal interval of consecutive integers which it contains: $$ \mu(N) = \max \{ y - x : [x,y) \subset U_N\} $$ $\mu$ can also be defined e...

For a set $\mathcal{X}$ to be countable, it should be either finite or countably infinite. Now, to show that a set is countable, we can apply a function $f$ that will map it to another set, say $\mathcal{Y}$. If we show that this function is one-to-one, then it's sufficient to prove that $\mathca...