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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Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
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196 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
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112 views

What is the smallest integer $n$ for which $\theta(n) > n$?

What is the smallest integer $n$ for which $\theta(n) > n$? Here $\theta(x) = \sum_{p \leq x} \log p$. I googled around, checked some likely textbooks, and ran a program for $n \leq 10^7$, but ...
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275 views

Estimation for logarithm of Riemann zeta function

Let $\sigma >1-\dfrac{c}{2\log(|t|+3)},|t|>7/8,$ where $c$ is constant from Theorem about region without zeros of Riemann zeta function. Using the fact that $$\log \zeta(s) - \log \zeta(s_1) = ...
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131 views

Solutions to simultaneous excluded congruences

I'm interested in the smallest solution to a family of "excluded congruences." To be precise, let $p_1 < \ldots< p_k$ be a sequence of primes and consider the constraints $$ x \not\equiv a_1 ...
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182 views

Fastest Primality test using $N-1$ Factorization?

If $N-1$ could be factored easily with several small prime factors, then what is the fastest way to check $N$ for primality? Updated I'm aware of Pocklington primility test which is not good for ...
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157 views

Extensions of Mixed Hodge Structures

The analogy on the front page of this paper by Bloch and Kriz seems like it's going to be lovely, but I don't get it, because I don't know how to view a torsor for $\mathbb{Q}(1)$ as an extension of ...
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143 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
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112 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
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253 views

Weak Dirichlet's theorem for powers of primes

Let $p$ be a prime, and let $m$ be an integer coprime to $p$. Then fix a natural number $k>0$. Is there any result that is simpler than the full Dirichlet's theorem that proves the existence of ...
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184 views

$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
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85 views

Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$

I'm looking for composite $n$ such that $$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$ Are there only finitely many? Can this be proved? This is Sloane's A070037 but there's not much information in the ...
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164 views

Proof of infinitude of primes whose reversal in base 10 is also prime

Is there any proof of infinitude of http://oeis.org/A007500 primes. If you want to generate them here is trivial and naive python program. ...
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541 views

Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
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175 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
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296 views

Prime zeta definition, multiplication by zero

Wikipedia has a page about the prime zeta function which is defined as follows: $$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$ I entered this additional definition: Define a sequence: ...
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146 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
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192 views

A subtle relationship from class field theory

Recently, I consider a problem: Let $E/F$ is a Galois extension of number field, denote the idele group of $E$ (resp $F$) by $I_E$ (resp $I_F$). There is a homomorphism induced by norm map $N_{E/F}$: ...
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62 views

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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65 views

Proving the decomposition $N = a^a b^b$ is unique or not

Suppose $a,b,c,d$ be natural numbers. If $a \ge b > 0$, $c \ge d > 0$, $a^a b^b = c^c d^d$, then $a = c$ and $b = d$. For example, can we find another decomposition of $N$ when $N = 8^8 4^4$ ? ...
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44 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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38 views

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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51 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 ...
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51 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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66 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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64 views

Equation $(x+y\sqrt{2})^6+(u+v\sqrt{2})^6=5+7\sqrt{2}$ in $\mathbb{Q}$

Solve equation $$(x+y\sqrt{2})^6+(u+v\sqrt{2})^6=5+7\sqrt{2}$$ for $x,y,u,v$ rationals. Sorry but I have not any idea.
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42 views

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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113 views

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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52 views

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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100 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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108 views

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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53 views

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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316 views

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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55 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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38 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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99 views

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 ...
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64 views

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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130 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 ...
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114 views

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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42 views

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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51 views

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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140 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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58 views

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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86 views

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*} ...
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40 views

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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64 views

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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30 views

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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75 views

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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38 views

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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44 views

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 ...