Articles of number theory

Elementary proof of Catalan's conjecture – valid or not?

Many months ago, I have found this paper: 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 […]

Has the 3×3 magic square of all squares entries been solved?

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?

Is there any formula to calculate the number of different Pythagorean triangle with a hypotenuse length $n$, using its prime decomposition?

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 […]

Degree $2$ nilpotent matrices with non-zero product

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 […]

Prove that function is continuous in all irrational points

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 […]

Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?

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 […]

The set of exponential primes

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 […]

Finding Extra Condition for a function to satisfy $f(n)=n$

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 […]

Cramér's Model – “The Prime Numbers and Their Distribution” – Part 2

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.

Numbers which are not the sum of distinct squares

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. $30$. $$30 = 1^2 + 2^2 + 5^2 = 1^2 +2^2 + 3^2 + 4^2$$ […]