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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if $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$?

is it true that for any $k,l\in\{2,3,4,\dots\}$, $k\neq l$, if $x\in\mathbb{R}$ satisfies $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$? This is a generalisation of if $x^3-x\...
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Interesting properties of a mathematical number theory game

The game, which is purely recreational, goes as follows: Starting out with 1, you can employ any of two different generation rules: You can multiply by 3 You can divide by two, rounding up (e.g. 3 ...
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What is the smallest prime factor of the number $14^{14^{14}}+13\ $?

What is the smallest prime factor of the number $$N\ :=\ 14^{14^{14}}+13\ ?$$ The number of digits of $N$ is $12,735,782,555,419,983$ (The number of digits of $N$ has itself $17$ digits). The first ...
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49 views

The group defined by Gauss's definition of composition of forms

In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. ...
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Evaluating a double sum involving prime numbers

Evaluate $$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{n}{(p-1)p^k} \right\}$$ where $\{ x\}$ denotes the fractional part of $x$, and $p \leq n$ denotes all ...
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53 views

Characteristic and minimal polynomials

Given $A\in\Bbb Z^{n\times n}$, is it possible to find characteristic and minimal polynomials of $A$ by chinese remainder theorem if we know characteristic and minimal polynomials respectively of $A\...
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77 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
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70 views

Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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71 views

Finite extensions of $\mathbb{Q}((t))$

Let $K$ be a number field. Is every finite extension of $K((t))$ of the form $L((\pi^{1/e}))$, where $L/K$ is finite and $\pi = a_1t + a_2t^2 + \cdots$ for some $a_i\in L$? Is every finite flat ...
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Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P ∈ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
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Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
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103 views

Perfect Prime $4D$-Cube

$\color{gray}{\mbox{ I don't want to overflow/burden you with numerous rigorous definitions...}}$ So, definition by the images:      Perfect Prime Cube:   $(30; 5, 22, ...
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Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
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Classifying Diophantine Equations

Take a given Diophantine equation. Chances are, we can't find any solutions. But if it's an equation of a certain form, we may get lucky and may be able not only to find a solution, but be able to ...
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What are the next primes in this sequence?

Define $z_k$ to be the smallest number $z$, such that $$z\equiv \phi(\phi(p))\pmod p$$ for every prime $p\le k$. We can assume, that $k$ itself is prime. The first few numbers are $z_2=z_3=1$ , $...
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LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \frac{\text{lcm}(1, 2, ... n+1)}{n+1}$$ 1) Is there any method to find $...
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60 views

Composite numbers $n$, such that every $a$ with $1<a<n-1$ is a witness

A Carmichael number $n$ has the property that $$a^{n-1}\equiv 1\ (\ mod\ n\ )$$ for every $a$ with $(a,n)=1$. I wonder, which numbers have the converse property : For which composite numbers $...
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45 views

De Rham-Etale comparison isomorphism for elliptic curves

I can't find anywhere a proof of the following comparison isomorphishm: $$H^1_{dR}(E)\otimes \mathbb{C}=H^1_{et}(E)\otimes \mathbb{C}$$ where $E$ is an elliptic curve over $\mathbb{C}$. Any ...
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Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$.

Find all $x,y\in \mathbb{N}$ such that: $2^x+17=y^2$. Its easy to find that $x=6$ is the only even value for $x$, the others have to be odd. One more thing is that we get $y^2 \equiv 19 \pmod p$, ...
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Tate curve and action of inertia group

I read the answers to this question Clarifying a comment of Serre. However I miss a passage of the second answer and since I can't comment there I have should post a new question. I don't understand ...
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132 views

Diophantine equation: $13^x+3=y^2$

$x,y$ are positive integers. $$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore $y+\...
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Connections between Fibonacci and natural numbers

Here are some known facts about the Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem : For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of ...
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Motivation behind the Archimedean norm on number fields .

It is easy to justify the use of non-archimedean norms on number fields from an "inside view" as arising from the prime ideals and are therefore clearly useful a priori. However, it seems to me that ...
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Which permutations of $\mathbb{C}$ commute with the Riemann zeta function?

I'm trying to figure out whether the permutations of $\mathbb{C}$ which commute with the Riemann $\zeta$ function are necessarily continuous or not. Obviously both the identity and the complex ...
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146 views

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and $$a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c$$ for all $n\ge 1$. Prove that for each integer $n \ge 2$ there ...
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Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
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Special Case of Composite mersenne number mod p

We want to investigate if a composite mersenne number $p|2^{qb}-1$ where $p\nmid2^q-1$ ,$q,p$ are primes, $p=1+6qb,\ qb\equiv1(mod64) $ and $b$ is an odd number. In general for $$\begin{align*} x^n\...
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Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
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Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, (...
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The copy-problem : Does any block of digits appear at least twice?

Suppose, $N$ random digits have been generated. Let $X$ be the largest natural number with the following property : There are natural numbers $i$ and $j$ with $i+X-1<j$ , such that the digits $i$ ...
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Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for ...
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Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of $S$...
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Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\...
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Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using Zsigmondy, ...
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81 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If $4^p-1=3^k\cdot ...
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Exploiting a crypto backdoor based on a polynomial

At a capture-the-flag competition during the weekend, there was a task that involved the following polynomial over the field $F = \mathbb{F}_P$ of integers modulo $P = 571787215471557516425591$ (yes, ...
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A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
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Question about exponentiation

Consider exponentiation on the natural numbers. Suppose we have an equation involving only the variable $x$, where there are $m$ $x$'s on one side and $n$ $x$'s on another side where $m$ and $n$ are ...
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The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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Characterization of Extended Lucky Numbers

The Lucky Numbers is a sieve where one starts out with the positive integers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ... and then eliminate ...
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Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
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Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
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Not using Jacobi symbol how to prove For all positive integer $n>1$ $2^n - 1 \not | 3^n-1$?

There is a proof: if $n$ is even,then $3|2^n-1$ but $3\not|\;3^n-1$,It is correct; if $n$ is odd,suppose $2^n-1|3^n-1$,then $3^n \equiv 1(\mod 2^n-1)$,then $$\left(\frac{3^n}{2^n-1}\right)=\left(\...
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More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=3m+1\tag1$$ and the cubic, $$x^3+x^2-mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac {s+1+3\,n}{...
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Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of $Homeo(\mathbb{R})...
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Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv \left(\frac{p+1}{4}\right)+\left(\frac{p+1}{4}\...