Articles of number theory

Can $\pi$ be rational in some base radix

I am from a physics background and my mathematics is not very good, so pardon my insolence with the question. Editing based on the comments : We know that $\pi$ in decimal (i.e. base 10) is transcendental. Is it possible to have a radix base in which $\pi$ can be rational in that base ? […]

Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?

I was watching this video on YouTube where it is told (at 6:26) that $2^{16} = 65536$ has no powers of $2$ in it when represented in base-$10$. Then he – I think as a joke – says “Go on, find another power of $2$ that doesn’t have a power of $2$ digit within it. […]

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan’s octic continued fraction which I discovered using certain three term relations and algebraic manipulations. Given $$\big(u(2\tau)\big)^2=\cfrac{2\,q^{1/2}}{1-q+\cfrac{q(1+q)^2}{1-q^3+\cfrac{q^2(1+q^2)^2}{1-q^5+\cfrac{q^3(1+q^3)^2}{1-q^7+\ddots}}}}$$ then by using the well known special value $$\big(u(i)\big)^2= \sqrt{2}-1$$ which was first found by Srinivasa Ramanujan in his […]

Efficiently finding two squares which sum to a prime

The web is littered with any number of pages (example) giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no such pairs for primes congruent to 3 mod 4). However, none of the stuff I’ve read on […]

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$

How to show $e^{e^{e^{79}}}$ is not an integer

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? After several years, it appears this is an open problem. As a non-number theorist, I had assumed there would be known results that would answer […]

Any odd number is of form $a+b$ where $a^2+b^2$ is prime

This conjecture is tested for all odd natural numbers less than $10^8$: If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$. $\mathbb P$ is the set of prime numbers. I wish help with counterexamples, heuristics or a proof. Addendum: For odd $n$, $159<n<50,000$, there are […]

Show that $n$ does not divide $2^n – 1$ where $n$ is an integer greater than $1$?

This question already has an answer here: For $n \geq 2$, show that $n \nmid 2^{n}-1$ 2 answers

Theorem on natural density

In this answer to a question about a series, a theorem was stated: if $A= \\{a_i \\}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the natural density of the set. My background in number theory is basically zero and all my attempts to prove this […]

Chinese Remainder theorem with non-pairwise coprime moduli

Let $n_1,…,n_k \in \mathbb{N}$ and let $a_1,…,a_k \in \mathbb{Z}$. How to prove the following version of the Chinese remainder theorem (see here): There exists a $x \in \mathbb{Z}$ satisfying system of equations: $$x=a_1 \pmod {n_1}$$ $$x=a_2 \pmod {n_2}$$ $$\ldots$$ $$x=a_k \pmod{n_k}$$ if and only if $a_i=a_j \pmod{\gcd(n_i,n_j)}$ for all $i,j=1,…,k$? If numbers $n_i$, for $i=1,…,k$, […]