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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Prove a property of the divisor function

Let $q$ be an odd composite integer and $\sigma(q)$ the sum of the positive divisors of $q$. For what $q$ is it true that $$(\sigma(q)-q) \mid (q-1) \;?$$ If $q$ is prime, it is clear that it is ...
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Problem following detail of Szemeredi's original proof that sets of positive upper density have arbitrarily long arithmetic progressions

Here is a link to Szemeredi's original proof of Szemeredi's theorem: http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf I am stuck on Fact 6 (page 212) because I don't follow the "induction" ...
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$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
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Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. As usual, $\sigma_k(n) =\sum_{d\mid n} d^k$. This identity is ...
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112 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
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News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
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41 views

Useful restrictions for prime factors of a sum of two powers with coprime exponents

Every prime factor of a number of the form $$a^2+b^2$$ with gcd(a,b)=1 has -1 as a quadratic residue. Does this work only for exponents with a common factor, or are there useful restrictions also ...
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polynomials and functions on $\mathbb{Z}/n\mathbb{Z}$

My general question is How is the set of all polynomial functions on $\mathbb{Z}/n\mathbb{Z}$ structured? What is the number of such functions? How, given a function, one can recognize that it is ...
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An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
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Are the Fourier coefficients of a new form real?

Let $f\in S^\text{new}_k(Γ_0(N))$ be a $\text{newform}$ . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?
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$\frac{p^p-1}{p-1}, \frac{p^p+1}{p+1} $ cannot be prime power at the same time

$p\gt3$ is a prime, then the two numbers $$\frac{p^p-1}{p-1}, \; \frac{p^p+1}{p+1} $$ cannot be prime power at the same time I have no clue about it. Could anyone help me? Thanks a lot.
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168 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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79 views

Transcendental numbers involving primes?

Is the prime zeta function value $$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$ a transcendental number ? What about the following sum ...
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Ramification index and residue class degree under completion

I've got a problem in proving something written at page 111 of the book "Algebraic Number Theory" by A. Fröhlich and M. J. Taylor. This is the setting. Let $\mathfrak{o}$ be a Dedekind domain with ...
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76 views

Squarefree numbers

From the Wolfram MathWorld page on squarefree numbers, "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer." ...
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48 views

$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
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145 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
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58 views

How prove this $\frac{a_{1}+a_{2}+a_{3}+\cdots+a_{p^2-q^2}}{p^2-q^2}=p^2-q^2+1$

Nice Question: let $p,q\in N^{+}$,and $p>\sqrt{2}q\ge \sqrt{2}$ show that: there are exsit $a_{1},a_{2},a_{3},\cdots,a_{p^2-q^2}$ such $a_{i},(i=1,2,\cdots,p^2-q^2)$ are all positive square ...
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special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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39 views

Equations with operations as variables

There are number puzzles which go like this: given 2_2_1=5. Insert operations (addition and multiplication) to make the equation valid. Solution: (+,+) or ($\cdot$,+). My question is: does anybody ...
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331 views

Totient summatory function

Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a ...
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Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
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107 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
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68 views

Remark in “Multiplicative Number Theory” by Davenport

On page 9 in "Multiplicative Number Theory" by Davenport, he remarks that "It is a remarkable fact that no one has yet given a simple and direct proof that the value of the finite sum in (7) is ...
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69 views

Conditions for polynomial $f$ such that $f(n) \in \mathbb{N}$ for enough $n \in \mathbb{N}^+$ implies $f$ has rational coefficients

This question is suggested by this one: prove: coefficients of $f(x)$ are rational numbers What are the weakest sufficient conditions and strongest necessary conditions on a set of positive integers ...
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116 views

How to solve $a^x=x\,(\text{mod } 10^{12})$

For given $1< a< 50001$, I want to find $x$ for which $a^x=x ~(\text{mod }10^{12})$. My idea is to find $x_1$: $a^{x_1} = x_1~ (\text{mod } 2^{12})$ and $x_2$: $a^{x_2} = x_2 ~(\text{mod } ...
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Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
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Symmetry in the Sum of Factors

The factors of 72 are 1,2,3,4,6,8,9,12,18,24,36,72. The factors can be organized as such $$m=1,2,3,4,6,8$$ $$n=72,36,24,18,12,9$$ Knowing the symmetric properties and that the 12 factors exist. Pair ...
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A computation modulo 1223

I want to compute $48^{306}$ modulo $1223$. I can't use a calculator, hence I tried to simplify something. I have $306=2^8+2^5+2^4+2$, thus $48^{306}=48^{2^8+2^5+2^4+2}=48^{2^8}\cdot 48^{2^5}\cdot ...
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125 views

Proving that the finite sum of the each reciprocal of any sequence of integers with common difference is not an integer.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$. A week ago, I found the following question in a book: Prove that ...
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Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
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161 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
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102 views

$\pi$, disjunctive numbers, and finite sequences of given length

It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion of $\pi$. Of course, every sequence ...
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63 views

Why does there exist a totally positive element?

Given a totally real number field $K_0$ and its totally imaginary quadratic extension $K$. Does there exist an element $\psi\in O_K$ such that $-\psi^2$ is totally positive in $K_0$? Why?
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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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51 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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64 views

About amicable numbers

My question might seem wrong but I will explain it now: We will define $\phi(a)$ as the sum of divisors of $a$ except $a$. A pair $(a,b)$ is an amicable pair $\iff ((\phi(a) = b) \land (\phi(b) = ...
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A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
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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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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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213 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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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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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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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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73 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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187 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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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. ...