Many months ago, I have found this paper: https://hal.archives-ouvertes.fr/file/index/docid/678031/filename/ethcatalan2.pdf which is supposed to give an elementary proof of Catalan’s conjecture, a.k.a. Mihailescu theorem. Back then I wasn’t really able to follow the paper (many transitions involve ad hoc defined numbers and their definitions are all over the paper) but did not question its correctness. Today […]

It is my understanding that it has not yet been determined if it is possible to construct a $3$x$3$ magic square where all the entries are squares of integers. Is this correct? Has any published work been done on this problem?

Lets define $N(n)$ to be the number of different Pythagorean triangles with hypotenuse length equal to $n$. One would see that for prime number $p$, where $p=2$ or $p\equiv 3 \pmod 4$, $N(p)=0$ also $N(p^k)=0$. e.g. $N(2)=N(4)=N(8)=N(16)=0$ But for prime number $p$, where $p\equiv 1 \pmod 4$, $N(p)=1$ and $N(p^k)=k$. e.g. $N(5)=N(13)=N(17)=1$ and $N(25)=2$ and […]

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and (2) every product of the permutations of $M_i$ doesn’t vanish: $\prod_{i} M_i \ne 0$? If it is possible, $M_i$ better be as small […]

We have a function $f: \mathbb R \to \mathbb R$, $x \mapsto \frac{1}{n}$ when $x \in \mathbb Q, x = \frac{z}{n}, z \in \mathbb Z, n \in \mathbb N, z \text{ and } n$ are coprime, $x \mapsto 0$ when $x \notin \mathbb Q$. We need to show that $f$ is continuos in any $x […]

A positive integer $N$ is said to be $k$-multiperfect if $$\sigma(N) = kN$$ where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer. (The case $k = 2$ reduces to the original notion of perfect numbers.) Now my question is the following: Are all known $k$-multiperfect numbers (for $k […]

Consider a set of integers $Q$ such that the set of all positive integers $\mathbb{Z}$ is equivalent to the span of ever possible power tower $$a_1^{a_2^{\ldots a_N}}$$ involving $a_i \in Q$. In simpler terms. Take the integers, remove all square numbers, cube numbers, fourth powers, fifth powers, etc… And this remaining set is $Q$. What […]

Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions: If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$. $f$ strictly increasing. $f(2) = 2$. Find a 4th condition such that the result will be that $f(n)$ must equal $n$ for any natural number $n$. (Of course all the conditions […]

Following a previous question (here you’ll find an introduction): The book states that using the convergence of the binomial distribution towards the Poisson distribution, it’s easy to show that $$|\{x\le\xi:\pi_S(x+\lambda \log x)-\pi_S(x)=k\}|\sim\xi\mathrm e^{-\lambda} \frac{\lambda^k}{k!}\quad(\xi\to\infty)$$ holds almost surely. I couldn’t prove this.

We are defining square factorization as representation a positive natural number as sum of squares of different positive, integer numbers. For example $5 = 1^2 +2^2$ and $5$ has no more representation. But one number can possess more representations, eg. $30$. $$30 = 1^2 + 2^2 + 5^2 = 1^2 +2^2 + 3^2 + 4^2$$ […]

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