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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Applications of advanced number theory to other areas of math

In a recent conversation with a friend, I was discussing the fact that out of all of the fields of math, number theory seems to be among those that apply ideas from a large number of different fields. For example, modern results in number theory…
Dorebell
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Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite nested radical…
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Number of common divisors between two given numbers

How can I compute the number of common divisors of two naturals ? For example,if we consider (12,24) the answer is 6 i.e {1,12,2,6,3,4}. EDIT : I got an answer here.The solution boils down to finding the number of divisor of the GCD of the two…
Quixotic
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Continued Fraction expansion of $\tan(1)$

Prove that the continued fraction of $\tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]$. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a recurrence relation, but that didn't seem to…
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Irrationality of Two Series

Show that if the integers $1
Kou
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27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acting…
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Decimal form of irrational numbers

In the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for irrational numbers ? In other words, for an…
user165633
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If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude that $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an…
Clin
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If $(a_n)$ is increasing and $\lim_{n\to\infty}\frac{a_{n+1}}{a_1\dotsb a_n}=+\infty$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{a_1a_2\dotsb a_n}=+\infty$$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an irrational number proof by contradiction? …
Clin
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Finding 1000th 5-smooth number

Smooth numbers are natural numbers that are products of only small prime numbers. They have some applications in cryptography. A number is 5-smooth if its only prime factors are $2,3$ or $5$. Example: $$1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15,…
VividD
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Equivalence to the prime number theorem

I was just reading this question and answer: How will this equation imply PNT and it raised a whole new question: Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} \frac{\Lambda(n)}{n}=\log x-\gamma +o(1),$$ where gamma…
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Why is $(x^2-2)/(2y^2+3)$ never an integer for any integers $x$ and $y$?

I've started a little reading on quadratic reciprocity, and a reason for this has eluded me. Here's a little of what I came up with so far. I decided I want to show that for all primes $p$, if $p|x^2-2$, then $p$ does not divide $2y^2+3$. Then, by…
yunone
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General formula to obtain triangular-square numbers

I am trying to find a general formula for triangular square numbers. I have calculated some terms of the triangular-square sequence ($TS_n$): $TS_n=$1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 1882672131025, 63955431761796,…
RayQuang
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Prove that if $n(a^2+b^2+c^2)=abc$ then $2\mid n$

Is it true that if $n\in\mathbb N$ and the diophantine equation $$n(a^2+b^2+c^2)=abc,\\(a,b)=(b,c)=(c,a)=1\tag1$$ has positive integer solutions $a,b,c$, then $2\mid n$? I can prove that $3\mid n:$ 1) If $3\not\mid abc$ then $3\mid…
lsr314
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Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book

My question is about the end of the proof of theorem 1.1, in page 27. Namely, it is stated that whenever we have a multiplicative function $f:\mathbb{N} \to \mathbb{C},$ let the sequence $\Lambda_{f}(n)$ be defined via the Dirichlet series…
Mr. No
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