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 algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

Number theory is concerned with the study of natural numbers. One of the main subjects is studying the behavior of prime numbers.

We know that by the prime number theorem, the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. Another good approximation is $\operatorname{li}(x)$. Despite these estimates, we don't know much about the maximal prime gaps. The weaker conjectures, such as Legendre's conjecture, Andrica's conjecture and Opperman's conjecture, imply a gap of $O\left(\sqrt{p}\right)$. Stronger conjectures even imply a gap of $O(\ln^2(p))$. The Riemann Hypothesis implies a gap of $O\left(\sqrt{p} \ln(p)\right)$, though proving this is not sufficient to show the RH. The minimal gap is also a subject of research. It has been shown that gaps smaller than or equal to $246$ occur infinitely often. It is conjectured that gaps equal to $2$ occur infinitely often. This is known as the twin prime conjecture.

Another subject in number theory are Diophantine equations, which are polynomial equations in more than one variable, where variables are integer valued. Some equations can be solved by considering terms modulo some number or by considering divisors, prime factors or the number of divisors. Other equations, such as Fermat's Last Theorem, are much harder, and are or were famous open problems. Recent progress usually uses algebraic number theory and the related elliptic curves.

Another subject is the study of number theoretic functions, most notably $\tau(n)$, the number of divisiors of $n$, $\sigma(n)$, the sum of divisors of $n$ and $\varphi(n)$, the Euler-phi function, the number of numbers smaller than $n$ coprime with $n$.

For questions on elementary topics such as congruences, linear Diophantine equations, greatest common divisors, quadratic and power residues, primitive roots, please use the tag. This tag is for more advanced topics such as 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 algebraic 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?

$\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever…
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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, \begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align} My question is if whoever invented the decimal system had chosen $9$…
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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 $3$ and add $1$ to obtain $3n + 1$. Repeat the…
Dan Brumleve
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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 $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from…
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Books on Number Theory for Layman

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, it appears this is an open problem. As a non-number…
Carl Mummert
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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 prime before $7$. Similarly, the prime…
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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 factors. That made me realize that $64$ would be an…
Brian Rothstein
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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 him? The algebraist argues that the real numbers…
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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 slightly, $$ a^{n - 1} \equiv a \pmod{n}, $$ the values of…
Tavian Barnes
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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 $\sqrt{n^2} = n$ for any positive integer $n$. …
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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 tempted to multiply, resulting in $\frac{8}{3}$. My questions pertain to…
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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 $1,$ and show that $n < x < n+1$ for some integer $n$. …
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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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ with $a_i \in \mathbb{Q}$. However, the ring of…
Eins Null
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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 totient function to each term, the equation…
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