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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Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
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39 views

$\sum_{a^2<p\leq (a+1)^2}p$ Summation of primes

$$\sum_{a^2<p\leq (a+1)^2}p$$ where p is prime. Are there some known bounds for this sum?
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150 views

Number of nonnegative solutions of linear diophantine inequality

Given inequality $Ax + By \le C$, where $A, B, C$ are integers, $A$ and $B$ are coprime and $C < AB$. I need to find number of non-negative integer solutions of it. Is there exists algorithm which ...
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117 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
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141 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
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31 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
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52 views

Artin-Schreier Question from Corps Locaux

I have a question from Serre's book "Corps Locaux", namely question 5a in section 2 of chapter IV. It is as follows: "Let $e_K$ be the absolute ramification index of K, and let n be a positive ...
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392 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
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178 views

Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
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128 views

Number of Solutions to a Diophantine Equation

I am asked the following: Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. I checked that for $y^p=x^2+2$, the same method for $y^3=x^2+2$ works ...
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84 views

Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for $y\in\...
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135 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
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95 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: $$\frac{1}{x}\left(\sum_{k=1}^xM(k)...
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80 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum $$\...
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135 views

Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...
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73 views

$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ 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 $P_{K,1}(\...
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59 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then $\...
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112 views

How find this equation all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$ My try: If $n>1$ is odd, then note $$\left(n^{\frac{n+1}{2}}\right)^2<n^{n+1}+n-1<\left(n^{\frac{n+1}{2}}+1\right)^2$$ so ...
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61 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
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149 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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50 views

List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
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102 views

How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&...
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139 views

References request: Ramanujan's tau function.

References request: Ramanujan's tau function. Let $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n$, $q=e^{2\pi i z}$. How can one show the following using representation ...
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77 views

Intersection between OEIS-A050808 and OEIS-A080053

Is there any result on the intersection between A050808 and A080053: OEIS-A080053 = 1, 2, 4, 5, 6, 10,... (Exp(n) is further from an integer than any previous exp(k)) OEIS-A050808 = 1, 2, 18, 50,...
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119 views

Quadratic equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $\,f(x)=ax^{2}+bx+c\,$ in $\,\mathbb{Z}/ n\mathbb{Z},\,$ we know that this quadratic equation $\,f(x)=0\,$ has ...
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118 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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68 views

Is there always a prime number in the form of $4k+1$ between $[n, 2n]$ for every large enough $n$?

Is there always a prime number in the form of $4k+1$ between $[n, 2n]$ for every large enough $n$? I guess it is known as a classical result. Is there any reference for it? Thanks!
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72 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
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67 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
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452 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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114 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
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123 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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129 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
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136 views

Adelic lattices

Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
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162 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
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420 views

The greatest prime factor of $6n+1$

Let $p(n)$ be the greatest prime factor of $n$. Denote $a_1(n)=p(6n+1),a_{k+1}(n)=p(6a_k(n)+1)).$ Is it true that for $\forall n\in \mathbb N,$ $\exists c,t \in \mathbb N^{+}:k>c\implies a_{k+t}(n)...
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93 views

Surjective map in Galois cohomology?

Let $p$ be a prime number, $\mathbb{F}_p$ the field with $p$ element and $\omega$ the mod $p$ cyclotomic character. Let $K$ be a finite extension of $\mathbb{Q}_p$ (the field of $p$-adic numbers) and ...
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657 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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98 views

Linear independence of $\cos(n\theta)$

I was trying to see if the cosines of the (certain) integer multiples of a certain angle were linearly independent over $\mathbf{Q}$. In particular I was looking at when $\theta = \frac{2\pi}{p(p+1)}...
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100 views

Bounds for multi-dimensional Kloosterman Sums

I'm looking for a general bound (in terms of $p$) for the Kloosterman sum, working in $\mathbb{F}_{p}$, $$\sum\limits_{x_{1} \dots \ x_{n} = a} \psi(x_{1} + \dots + x_{n})$$ for $\psi$ a nontrivial ...
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195 views

possible prime factors of $4^{444}+3$

I have not factored the number $4^{444}+3$ yet. I wonder, though, if there are restrictions for possible prime factors p. The only obvious restriction is, that -3 must be a quadratic residue of p. ...
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268 views

Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is coprime to $10$,such that the period of the decimal expansion of $1/n$ is a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If $n-1=2^xc$ or $...
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97 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
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117 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
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119 views

Characterization of quadratic polynomials over $\mathbb{Z}/p\mathbb{Z}$

This is a nice question which I'd like to share with everyone. Let $f:\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$ be a function s.t. for each $a \in (\mathbb{Z}/p\mathbb{Z})^{\times},$ the ...
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163 views

Primes dividing sequence

Define a sequence $a_n$ as such. $a_0=1$, $a_n=a_{n-1}+a\lfloor\frac{n}{3}\rfloor, \forall n\ge1$ Find all primes $p$ such that p divides infinitely many values of $a_i$. Edit: This is extension to p=...
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392 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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622 views

Defining the Riemann-Roch space of a divisor

I'm doing a course on elliptic curves. It starts with a bit of a crash course in algebraic geometry, giving statements alone. We were given the following definition The Riemann-Roch space of $D$ ...
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131 views

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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53 views

S-Unit equations in cyclotomic fields

By a Siegel's result, one knows that there exist only finitely many solutions of the equation: $$x+y=1$$ where the unknowns $x$ and $y$ are units in the ring of integers of a cyclotomic field. Do you ...