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For a given $n \in \mathbb{N}$, consider the algebraic function: $$ f_n : \mathbb{R}^{n+2} \rightarrow \mathbb{C} $$ $$ f_n(x,y,z_1,\dots,z_n) = x + iy $$ We're going to define a 'simple period' to be …

The automorphism group is strictly larger than $G$. Note that the automorphism group is a semidirect product of $G$ with the stabiliser of a single vertex, so it suffices to show that the group of aut …

Suppose we have such an $n$ where $P(n)$ is false. Now define $(a, b, c) := (1, n(n-2), (n-1)^2)$ and observe that these three numbers are pairwise coprime and satisfy $a + b = c$. Then we have: $$ \ …

Yes. I'll talk about why elliptic curve discrete log is harder than ordinary discrete log. Suppose we have $g, h$ and want to find $n$ such that $g^n = h$. The usual methods for solving the discrete …

As Stanley Yao Xiao commented, a definite answer to this question would be equivalent to solving an open problem. If we assume two reasonable conjectures, however, then the sum $\sigma$ of the recipro …

Another construction is to let $n \notin V$ if and only if $n$ begins with at least $\sqrt{\log{n}}$ consecutive '9's when written in decimal. This satisfies the stronger property that there is no non …

Leonhard Euler knew that the infinite product: $$ \prod_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right)^{-1} = \sum_{n=1}^{\infty} \frac{1}{n} $$ is divergent (and used this to prove the infinitud …

It's well known that the sum of the reciprocals of the primes below $n$ tends to $\log \log n + M$, where $M$ is a small constant (the Meissel-Mertens constant). That is to say: $$ \sum\limits_{\smal …

Let $\zeta$ be a primitive $2^k$th root of unity, and consider the ring $ \mathbb{Z}[\zeta, \frac{1}{2}] \subset \mathbb{C}$. Are there any elements of absolute value 1 other than the powers of $\zeta …

Yes, these are the only examples. Either $x$ is zero (your $(0, \pm 2)$ example) or not; assume the latter. Rewrite your equation as $(y - 2)(y + 2) = x^5$, and appeal to unique factorisation. Then $ …

I can prove that all sufficiently large integers are representable. Firstly, observe that there is a unique way, up to isomorphism, to choose three edges $p, q, r$ of the Petersen graph such that the …

It is possible to exploit the symmetries of the Petersen graph, together with the rearrangement inequality, to reduce the size of a brute-force search from $15!$ to $129729600$ (a $10080$-fold improve …

Commutativity is not necessary for the notion of primes. For instance, consider the Hurwitz integers, namely quaternions whose components are either all integers or all half-integers: $$ H = \{ a + b …

Definition: A weak prime gap is an integer which occurs infinitely often as the difference between two (not necessarily adjacent) primes. We'll show that the set of weak prime gaps has positive densi …

Before you can formulate your question precisely, you need a better notion of distance between two pictures of a set $S \subset \mathbb{C}$ (where in this case, $S$ is the union of Ford circles). If $ …