Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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How does Hildebrands proof of the prime number theorem via large sieve work?

How does the sieve inequality (I may not know the most general form) lead to the distribution of primes? To me, these concepts do not seem to be related. Can their connection be described in a ...
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58 views

General question about 'vieta jumping'

Suppose I want to prove that a variable posesses a certain property (e.g. is a square). For example if I wanted to prove that $x$ in $\frac{x^2+y^2+1}{xy} = k$ has the property of being a square (It ...
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84 views

How to prove that $n=2^k$?

Let two collection $\{a_{1},a_{2},\cdots,a_{n}\}\neq\{b_{1},b_{2},\cdots,b_{n}\}$ and $a_{i},b_{i}\in\Bbb Q$ for $i=1,2,\cdots,n $ be such that $$\{a_{i}+a_{j}\mid1\le i<j\le ...
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132 views

find the $ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$

let $x,y\in Z$,and find the equation :$$ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$$ all integer solution my idea: $$\Longleftrightarrow (y^3+xy-1)(x^2+x-y)-(x^3-xy+1)(y^2+x-y)=0$$ ...
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An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
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257 views

Origins of the Twin Prime Conjecture

The exciting new results by Zhang and others about bounds on the gaps between pairs of primes have been getting a fair amount of press, which is great! Some of them have gotten me wondering about the ...
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54 views

étale cohomology of valuation rings

Let $S$ be the spectrum of a discrete valuation ring (we can assume complete or henselian if necessary). Is it true that the étale cohomology group $H_{et}^2(S,\mathbb{Z})$ is zero?If not in general ...
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85 views

Integer values of the Riemann Zeta function

The when $s$ is real and greater than 1, the Riemann zeta function $\zeta(s)$ takes all finite positive value $> 1$. I am studying the values of $s$ for which $\zeta(s)$ is a positive integer. I ...
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120 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ...
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210 views

ECM on extremely big numbers?

Denote the following recursion $$a(0):=0 , a(n+1):=a(n)^2+1$$ I search a prime factor of a(89). There is none below 1,5*10^9. As the number itself is far too big, ECM cannot be used directly. But ...
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A challenging problem on prime uncertainty interval

I have a very challenging problem to solve, seeking for good advice; I have to make an intro in the first part and then comming to the problem. Theorem (1): In an interval between a prime $p$ and its ...
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74 views

About linear recurrence sequences

Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy \begin{eqnarray*} &&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\ ...
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203 views

primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$

I have trouble showing that primes of the form $p=8k+1, 8k+3$ can be expressed as $p=a^2+2b^2$. Thanks in advance.
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96 views

Prime norm ideals that are also principal

Landau's prime number theorem tells us asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X. I am interested in the the prime ideals with a prime norm. ...
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Use the ring ${\bf{O}}[\sqrt5]$ to show that $2$ is irreducible in $\mathbb{Z}[\sqrt5]$

I have this question: Define $${\bf{O}}[\sqrt5] = \{c_1 + c_2 \sqrt5 : c_1 + c_2 \in \mathbb{Z} \wedge c_1 - c_2 \in \mathbb{Z}\}.$$ This ring properly contains $\mathbb{Z}[\sqrt5]$. The ...
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202 views

Explicit Formula

In the explicit expression for $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ $ x^\rho$ denotes $x^{\mathrm{Re} \rho}$. I wanted to ...
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264 views

sum of digits in different bases

Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search?
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question on minimal polynomial

Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. Find the minimal polynomial of $\alpha^2+\beta$. My attempt to solution was this: i found a monic polynomial with integer ...
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312 views

Fermat Last Theorem for 13,17,19

I am asked to prove the Case I of Fermat’s Last Theorem for the exponents 13, 17 and 19. FLT states that the equation $x^n+y^n=z^n$ has no integer solutions for and $x,y,z$ different from zero. ...
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What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
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143 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
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Hilbert symbol over a ring

Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows: $$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
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Is this set dense on the positive half of the real line?

We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime. The ...
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124 views

Estimation on Primorial Influence

As you know, Primorial ($\#$) notion is defined as the product of first $n$ prime numbers. That is, $$ n\# = \prod_{i=1}^nP_i $$ An there is some effect named Primorial Influence (explained ...
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81 views

Prime clasfication by some constructive function

How to prove or justify the following: $$ f(g)= \frac1{1-g^2} \prod_{k=1}^{\infty} \left(\frac{\sin(\pi \frac gk)}{\pi \frac gk} \cdot \frac 1{1-\frac{g^2}{k^2}}\right), $$ The above statment can ...
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58 views

On CM Jacobians

I am looking for an example of a curve whose Jacobian is an abelian variety with complex multiplication by a non-abelian number field. Does anybody know such an example? Does it exist? Thanks
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107 views

Limit involving sums of the Von-Mangoldt function

Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem? $$ \lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) ...
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Number Theory Problem (On Fermat Little Theorem)

How to prove that the equation $x^2+5=y^3$ has no integer solutions? I have proved the case when $x$ is odd. I used the fact $x^2\equiv 1 \pmod 4$ but how would you do for even $x$: the mod 4 analysis ...
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Is this a known Pythagorean result?

Since the Pythagorean Theorem applies to right triangles, it can be stated: In triangle $ABC$, the length of whose sides are $a$, $b$ and $c$, $a^2+b^2=c^2$ if and only if $\cos(\angle C)=0$. I ...
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216 views

Sums of rows in the Pascal triangle

Assume for simplicity that when $k>n$ we have ${n\choose k}=0$. It is well-known that $\sum_k {n\choose 2k}=\sum_k {n\choose 2k+1}=2^{n-1}$ , i.e. the sum of the odd places in each row in Pascal's ...
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105 views

What is a cusp parameter?

I was reading this paper, and on the first page they define a cusp form as $$ f(z) = \sum_{n > -\alpha} a(n) e^{2\pi i (n + \alpha)z}. $$ Is this equivalent to the usual definition of a cusp form ...
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320 views

A different approach to the strong Goldbach conjecture?

Consider the set A of prime numbers $p_i$ such that $p_i+6$ is not prime (listed in OEIS 140555; see comments thereto). Let 'Goldbach representation' mean a pair of odd prime numbers which sum to a ...
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92 views

Products of primes of the form $an + b$

What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)? Surely (except for the trivial cases) they are of order strictly between that of he ...
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100 views

root of a unit in a real biquadratic field

Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in ...
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652 views

Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
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Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
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309 views

Sylvester's Theorem and Schur Theorem

I'll probably end up asking more programming questions on StackExchange forums than math questions, but I'll lead off with a math question. In my Number Theory class this past semester, I worked on a ...
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144 views

Champernowne-like squares, are there any?

I read about the Champernowne constant on Wikipedia a couple of days ago, and I got curious about something similar: is there some "Champernowne-like" number; that is, a concatenation of all numbers ...
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71 views

Determining the number of classes

How do can I determine all classes of ideals of $\mathbb{Z}[\sqrt{-104}]$? Or $\mathbb{Z}[\sqrt{-132}]$? (so a list of representatives and showing they are not equivalent, and and that we get all ...
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189 views

How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where ...
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If $0\leq \cdots \lt s'''\lt s''\lt s'\lt s$ and $s''=((s')^2-k)/s$, $s'''=((s'')^2-k)/s',\ldots$, then $k$ is a perfect square

This is an IMO problem from 1988, problem 6. The book does not provide a proof of this part and it is eluding me. Let $$\cdots \lt s''' \lt s'' \lt s' \lt s$$ all be ...
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Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
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342 views

Ramanujan Series

I have the following question involving the series of $1/\pi^3$: Can we find such expansions by using the one for $1/\pi^3$ with $1/\pi^4$ or $1/\pi^n$ etc? Note that $$\frac{1}{32}\sum_{n=0}^\infty ...
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60 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?
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317 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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Properties preserved under the “reversal” of a recurrence equation

Consider the recurrence equation $u_n = f(u_{n-1},\ldots, u_{n-k})\;,$ defined for $n=0,1,2\ldots$. If this is a linear recurrence and the coefficient function on $u_{n-k}$ is nonzero in ...
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188 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
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171 views

Hauptmoduls for modular curves

If I have a modular curve, how does one in general find a Hauptmodul for this curve?
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289 views

Rational independence

The specific question is the following: I am given a set of $[L/2]$ numbers $$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$$ where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ ...
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195 views

The discrete Fourier transform of a Dirichlet charachter

I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language. I am trying to find the ...