Math and stuff

Exercise from P.62 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Find the smallest number of marbles in a jar so that one remains if taken out 2, 3, 5 at a time, but none remain if taken out 11 at a time. The answer obviously is a...

Exercise from P.34 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Let $V$ be the set of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy the differential equation $f’’ - f = 0$. Show that $V$ is a vector space over $\mathbb{R}$ and, assuming its dimension is 2, find...

Exercise from P.28 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Find all of the rational points on the curve $x^n + y^n = 1 $ where $n$ is an integer, $n > 2$. It is obvious that $(\pm 1, 0), (0, \pm 1)$ are solutions when $n$...

Exercise from P.26 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Find a square-free congruent number not in the list above, and show all the work to obtain it. Based on the discussion in the book, we know that all primitive congruent numbers can be obtained by using...

Exercise from P.16 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Show that $x^2 + y^2 = 3$ has no rational solutions. Assume there is. Then there must exist $(x, y) = (a/c, b/c)$ where $a, b, c \in \mathbb{N}$ and $\text{gcd}(a, b, c) = 1$. Then $a^2...