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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Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational…
lhf
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What is an example of a sequence which "thins out" and is finite?

When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in if I ask them if there's infinitely many or not,…
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Mathematicians shocked(?) to find pattern in prime numbers

There is an interesting recent article "Mathematicians shocked to find pattern in "random" prime numbers" in New Scientist. (Don't you love math titles in the popular press? Compare to the source paper's Unexpected Biases in the Distribution of…
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Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$ For $r=p/q$…
mathlove
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Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a problem? (This is not homework - just a problem I…
Mario Carneiro
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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ The Riemann hypothesis asserts that all the…
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Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
thyrgle
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Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?

The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? This would be so easy if these points could be on…
Ahmed Amir
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Can someone please explain the Riemann Hypothesis to me... in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
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Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ the only positive integer value of $n$ such that…
Dave L. Renfro
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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millenium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even grasp the entire problem, but seeing the hypothesis and the other problems I can't help wonder: what is the practical…
Mythio
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n}…
Mats Granvik
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Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every 3rd term as positive, thus it would…
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How and why does Grothendieck's work provide tools to attack problems in number theory?

This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing. I have always been fascinated with Grothendieck and the way he did mathematics. I've heard Mochizuki's work on the abc conjecture heavily…
Arrow
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