# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### 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 ...
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### On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
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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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
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### How to solve this two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I can not progress anymore......
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### Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
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### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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### Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
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### Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (...
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Given a complex number \begin{aligned}\frac{z}{n}=x+iy\end{aligned} and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\tan\left(... 0answers 112 views ### Smallest Subset of$\mathbb{R}_{>0}$Closed under Typical Operations Let$S$denote the smallest subset of$\mathbb{R}_{>0}$which includes$1$, and is closed under addition, multiplication, reciprocation, and the function$x,y \in \mathbb{R}_{>0} \mapsto x^y.$... 0answers 230 views ### Continued fraction with prime reciprocal entries We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ... 0answers 178 views ### Product of primes mod n Let$n$be an odd composite number. I'm trying to compute $$f(n)=\prod_{n/2<p<n}p\pmod n$$ where$p$ranges over the primes in the indicated region. Can this be done (significantly) faster ... 0answers 131 views ### What other prime numbers have been ruled out as counterexamples to the Feit-Thompson conjecture? Given distinct primes$p$and$q$, $$\frac{p^q - 1}{p - 1}$$ is never a divisor of $$\frac{q^p - 1}{q - 1}.$$ Or so we believe. If$p = 2$, then it's clear that no odd prime$q$can make a ... 0answers 357 views ### A sequence that avoids both arithmetic and geometric progressions Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences ... 0answers 527 views ### Prime factor of$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below$8*10^9$Nevertheless, the given number has still ... 0answers 553 views ### Conjecture---Identity for Sieve of Eratosthenes collisions. Let $$\beta(n,k) = \max_{d \leq k}(d|n)$$ $$S(k)= \sum_{n=1}^{k!} \beta(n,k),$$$\hspace{20mm}$and $$T(k)=\# \{ ~i\cdot j~~\big|_{i=1}^k \big|_{j=1}^{k!} \}$$ Does $$S(k)=T(k)?$$ See OEIS ... 0answers 96 views ### 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 ... 0answers 188 views ### How did Hecke come up with Hecke-operators? I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,... 0answers 138 views ### Does the category of algebraically closed fields of characteristic p change when p changes? EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let \mathrm{ACF}_p denote the category of algebraically ... 0answers 534 views ### Find All x values where f(x) is Perfect Square Is there a formula, method or anyway to find all positive x integer values (if exists) such that f(x) is Perfect square where f(x) is a quadratic equation? For example if I have the following ... 0answers 130 views ### Can anyone improve on this work and find a closed form of \zeta(3)? This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ... 0answers 271 views ### Dividing the whole into a minimal amount of parts to equally distribute it between different groups. Suppose we have a finite amount of numbers x_1, x_2, ..., x_n (x_i\in\mathbb{N}) and an object that should be divided into parts in such a way that it can be without further dividing distributed ... 0answers 137 views ### Congruence properties of x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6? It is known that given primitive (co-prime) integer solutions to,$$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$then there is one x_i such that z^4-x_i^4 is divisible by d_4=5^4. Additionally, Ward ... 0answers 189 views ### Nested Radicals Involving Primes How do you evaluate \sqrt { 2+\sqrt { 3+\sqrt { 5+\sqrt { 7+\sqrt { 11+ \dots } } } } } ? This question appears to be rather difficult as there is no way to perfectly know what p_{ n } is , ... 0answers 179 views ### Class field theory for p-groups. (IV.6, exercise 3 from Neukirch's ANT.) I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let G be a profinite p-... 0answers 272 views ### If a^n-1 divides b^n-1 too often, then b=a^k I am looking for references to the following problem, which I saw a long time ago and I think is a well-known problem (maybe from IMO or American Mathematical Monthly), I hope to remember it correctly.... 0answers 193 views ### Proving that the number of integer solutions of x^2-Ny^2=1 is infinite I am trying to prove that the number of integer solutions of x^2-Ny^2=1 is infinite whenever N is a squarefree integer. For this I define norm of a+b\sqrt N=a^2-Nb^2. Now I prove that a+b \sqrt ... 0answers 254 views ### Irreducibility of cyclotomic polynomials via schemes A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ... 0answers 996 views ### Motivation for Weil pairing The Weil pairing$$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$for an elliptic curve is defined as follows. Let \phi:E\to E' be an isogeny of degree n and \hat\phi:E'\to E be the dual ... 0answers 450 views ### Cramér's Model - “The Prime Numbers and Their Distribution” - Part 1 I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ... 0answers 172 views ### Biggest powers NOT containing all digits. Let m>1 be a natural number with m \not\equiv 0 \pmod{10} Consider the powers m^n , for which there is at least one digit not occurring in the decimal representation. Is there a largest n ... 0answers 281 views ### When does a modular form satisfy a differential equation with rational coefficients? Given a modular form f of weight k for a congruence subgroup \Gamma, and a modular function t for \Gamma with t(i\infty)=0, we can form a function F such that F(t(z))=f(z) (at least ... 0answers 125 views ### Finding 1/x^2 + 1/x^3 + 1/x^5 + \dots The following function came up in my work:$$ f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots. $$Naturally, this converges for ... 0answers 128 views ### Alternating sums of numbers divisible by 7 Let x_1,x_2,x_3,x_4,x_5,x_6 be given integers, not divisible by 7. Prove that at least one of the expressions of the form$$\pm x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$$is divisible by 7, where ... 0answers 170 views ### Are my calculations of a new constant similar to Mill's constant based on \lfloor A^{2^{n}}\rfloor and Bertrand's postulate correct? As Wikipedia explains in number theory, Mills' constant is defined as: "The smallest positive real number A such that the floor function of the double exponential function \lfloor A^{3^{n}}\... 0answers 356 views ### Power Diophantine equation involving primes: (p+q)^q-p^q-q^q+1=n^{p-q} Suppose p and q are prime numbers, and n>1 is a positive integer. Find all solutions to the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$What I have tried: Obviously$p>q$... 0answers 112 views ### Is there a clever way to find a smaller number that produces the Euclidean algorithm of given length? Is there a simple way to tell if for a given$n$there is$m$such that the Euclidean algorithm on$n,m$runs for a given number of steps, and/or a way to find$m$efficiently (other than testing all$...
An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...