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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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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115 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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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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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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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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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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138 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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82 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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64 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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Cubic sum of Gauss integers

It's known that any Gauss integer can be represented as the sum of three Gauss integer squares. (See my another problem.) Let $A$ be a set, define $nA=\{x\mid x=a_1+a_2+\cdots a_k,a_i\in A,1\leq ...
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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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370 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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214 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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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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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. ...
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Making fermat's little theorem for composite numbers the ultimate test.

It is a programming question but mathematics has a major role to play in it. I have to find the largest prime less than a number $n$. Note that $n\leq10^{18}$. I can go for Fermat's Little Theorem ...
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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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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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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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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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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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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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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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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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How write Dirichlet character sums for the terms of the von Mangoldt function?

The way to separately write the terms of the von Mangoldt function $\Lambda$ as Dirichlet character sums seems to be: $$\Lambda (1) = \sum\limits_{n=1}^{\infty } \frac{(e^{\Lambda (1)} \chi ...
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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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264 views

Smooth numbers algorithm

I am trying to understand quadratic sieve algorithm and now I am thinking of the way to check if number is smooth over a factor base? For example, say I have number $n = 87463$. First,I find bound $B ...
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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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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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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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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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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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340 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 ...