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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 $ …

We prove that there are only finitely many such intervals. Suppose $[36n + 7, 36n + 29]$ is one such interval. Now, $36n + 10$ and $36n + 15$ cannot both be divisible by $5$ (since one must occur …

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 …

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 …

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 …

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 …

Yes, let $x$ be Chaitin's constant. Then both $x$ and $\gamma + x$ are uncomputable, therefore irrational.

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 …

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 …

We first run the algorithm to determine whether any solutions exist, by dovetailing an enumeration of integer $n$-tuples together with an exhaustive search of solutions modulo prime powers, halting if …

It depends on the model of computation you're using. If your memory slots can contain arbitrarily large integers (rather than single bits), and you can perform arithmetic operations in constant time, …

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 …

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 …

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 …

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 …