I'm interested in the following two complementary questions: Are there an infinite number of primes $p$ such that there exists some $1\le n < p$ with $2^n \equiv 3 \bmod p$? Are there an infinite number of primes $p$ such that for all $1\le n < p$, $2^n \not\equiv 3\bmod p$? A resolution to Art...

There are some related questions, e.g. here (but with no answer). I'm concerned about the proof of Stirling's formula in Rudin's PMA. I've spent a good portion of the day trying to figure out with what tools presented in the book so far I can best understand this. For a full overview of the part ...

I'm looking at the following definition of improper integrals, as well as their convergence: Definition of an Improper Integral of Type 1: If $\int_a^t f(x) \, dx$ exists for every number $t > a$, then $\int_a^\infty f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$ provided this limit exists...

Let $K$ be an algebraically closed field of characteristic $0$. I want to see that $K[s^2, st, t^2]$ is integrally closed. The following proof is given at $k[x^2,xy,y^2]$ is integrally closed : We have $\newcommand\quotient[2]{{^{\Large #1}}/{_{ \Large #2}}} R = K[s^2, st, t^2] \cong \quotient{K[...

I. Fibonacci and Lucas numbers $$F_n = 1,1,2,3,5,8,13,\dots\\[7pt] L_n = 2, 1, 3, 4, 7, 11, 18\dots$$ There is a relationship between two consecutive terms such that, $$M(x,y) = x^2+xy-y^2$$ $$M(F_n,\,F_{n+1}) = \pm1$$ $$M(L_n,\,L_{n+1}) = \pm5$$ with the sign depending on whether $n$ is odd or e...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function. We say that a point $x$ is periodic if it exists some $m >0 \in \mathbb{N}$ such that $f^m(x) = x$ where $f^m$ means $f$ composed with itself $m$ times. I'm trying to prove that if $f$ has no periodic points then for every $x$ the ...

A circle and an ellipse with foci $F1, F2$ lying inside it are given. Construct a chord $AB$ of the circle touching the ellipse and such that $AF_1F_2B$ is a cyclic quadrilateral. For this problem , i am interested in synthetic approach , allthough i tried to use coordinate to see if this tangen...

This problem is found in a past test. Let $n$ be a positive integer. Find all polynomials P with real coefficients such that $P(x^2 + x-n^2) = P(x)^2 +P(x)$ for all real numbers x. I think that we should be comparing coefficients of certain terms. If we start by noting the degrees of the polynomi...

Looking at almost 10 year old notes from a course on commutative algebra, I found the following exercise which I had apparantly not done back then: Task: Compute the spectrum of the ring $R = \mathbb{Z}[x,y]/(x(2+xy),y(2+xy))$, where $\deg(x) = 1$ and $\deg(y) = -1$. Out of curiosity I started do...

The title may be confusing, I'll try to clarify first. For me, the weakest set theory in general is the extension of classical predicate logic with equality by the axiom of extensionality. Let's call it basic set theory. Now let's try to extend the basic set theory by asserting the existence of t...