# Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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### Does $\pi$ contain all possible number combinations?

I came across the following image: Which states: $\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. ...
28k views

### Is '$10$' a magical number or I am missing something?

It's a hilarious witty joke that points out how every base is '$10$' in its base. Like, 2 = 10 (base 2) 8 = 10 (base 8) My question is if whoever invented the ...
72k views

### What is the importance of the Collatz conjecture?

I have been fascinated by the Collatz problem since I first heard about it in high school. Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by ...
35k views

### Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
101k views

### Books on Number Theory for Layman [closed]

Books on Number Theory for anyone who loves Mathematics?(Beginner to Advanced & just for someone who has a basic grasp of math)
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### 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, ...
23k views

### Are $14$ and $21$ the only “interesting” numbers?

The numbers $14$ and $21$ are quite interesting. The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$. Note that $3$ is the prime after $2$ and $5$ is the ...
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### Is $2048$ the highest power of $2$ with all even digits (base ten)?

I have a friend who turned $32$ recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being $32$ was pretty good since not only is it even, it also has no odd ...
8k views

### A variation of Fermat's little theorem in the form $a^{n-d}\equiv a$ (mod $p$).

Fermat's little theorem states that for $n$ prime, $$a^n \equiv a \pmod{n}.$$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
9k views

### How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
5k views

### What happens when we (incorrectly) make improper fractions proper again?

Many folks avoid the "mixed number" notation such as $4\frac{2}{3}$ due to its ambiguity. The example could mean "$4$ and two thirds", i.e. $4+\frac{2}{3}$, but one may also be ...
15k views

### Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than $1,$ ...
37k views

### How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly ...
5k views

### Pythagorean triples that “survive” Euler's totient function

Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple: \begin{equation} a^2 + b^2 = c^2. \tag{1}\label{1} \end{equation} Additionally, suppose that when you apply Euler's ...
6k views

### Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

I was looking at a list of primes. I noticed that $\frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge. This led me to try $\frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to ...
19k views

### The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using elementary ...
7k views

### Is there any mathematical reason for this “digit-repetition-show”?

The number $$\sqrt{308642}$$ has a crazy decimal representation : $$555.5555777777773333333511111102222222719999970133335210666544640008\cdots$$ Is there any mathematical reason for so many ...
2k views

### Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
22k views

### Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not ...
3k views

### Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
2k views

### Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
9k views

### A multiplication algorithm found in a book by Paul Erdős: how does it work?

I am trying to understand the following problem from Erdős and Surányi's Topics in the theory of numbers (Springer), chapter 1 ("Divisibility, the Fundamental Theorem of Number Theory"): We ...
4k views

### Least prime of the form $38^n+31$

I search the least n such that $$38^n+31$$ is prime. I checked the $n$ upto $3000$ and found none, so the least prime of that form must have more than $4000$ digits. I am content with a probable ...
6k views

37k views

### Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
2k views

### Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the first ...
11k views

### How can adding an infinite number of rationals yield an irrational number?

For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
### A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational
Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...