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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Question about the elements of a reduced residue system relative a primorial $p_n\#$

I've been dividing up the elements of reduced residue system relative a prime $p_n$ into congruence classes modulo $p_{n+1}$ and I noticed that each congruence class is represented. If $r$ = the ...
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54 views

How many distinct factors of $n$ are less than $x$?

For some (squarefree) integer $n$ and some integer $x$, I would like to find an expression that gives, for all $n$ and $x$, a good upper bound on the function $$f(n, x) = \sum_{d|n, d < x} 1$$ ...
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46 views

the different of finite extension of p-adic numbers

Let $F=\mathbb Q_p$ be the field of p-aidc numbers. Let $\xi_n$ be a $n$-th primitive root of unity. Now consider the finite extension of fields $K/F$ where $K=\mathbb Q_p(\xi_n)$. I want to find the ...
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68 views

Do there exist any cycles for these number sequences?

We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots ...
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66 views

Inertia field of a compositum.

My question is regarding composite field extensions. The problem is motivated but not explicitly stated in the chapter 4 exercises of Marcus' Number Fields, in particular ex. 31 c) We first provide ...
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43 views

A bound on the squares of primes

If $p_n$ is the $n$th prime, what is the best known upper bound on $m$ such that $p_n \cdot p_{m+n} < p_{n+1}^2$?
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61 views

For what values of $x$ will $ax^2+b$ be perfect squares?

I need help on the this. Suppose that $ax^2+b$ is given where $a, b\in \mathbb Z$. Can we determine all the values of $x\in \mathbb Z$ such that $ax^2+b$ will be a perfect square ? Please help me. ...
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29 views

<Reference Request> Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This has been cross-posted to MO.) Good day! I would like to request for references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor. ...
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98 views

Question regarding the prime factors of $2^{35} - 1$

Question regarding the prime factors of $2^{35} - 1$ I just wanted to make a few things clear; 1) It is true to state that this cannot be a Mersenne prime (A number of the form $2^r - 1$ where ...
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35 views

Number of digits $d$ in $d^k$

The title says it all really. For example, how many occurences of $6$ are there in $6^k$? It starts 6, 36, 216, ... so 1, 1, 1... The question can now be generalized into any digit or group of digits. ...
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Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
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100 views

Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
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153 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)$$ If $F_\color{red}1=0$, is there a rational solution to $7F_3x^4+7F_5x^2+F_7 = 0$? Then for $k=1,2,8$, ...
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74 views

Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers ...
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78 views

the Teichmüller character

Let $d \geq 2$ be an integer, $K$ a number field containing the $d$-th roots of unity $\mu_d(\mathbb{C})$ and $\mathfrak{p}$ a prime ideal of $K$ not dividing $d$. Let $\mathbb{F}_q$ be the residue ...
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80 views

Number of $n$-permutations for which ${\tau}^k = id$

I am curious about the formula(any closed form) for the number of $n$-permutations $\tau$ such that ${\tau}^{n-1} = id$. How about for the case ${\tau}^n = id$ ?
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Finding the sum of the coefficient squared of a polynomial

Given a polynomial $$f(x) = \sum_{k = 0}^n a_kx^n = a_0+a_1x+ \ldots + a_{n-1}x^{n-1}+a_nx^n $$ is it possible to find a general expression for $$ \sum_{k = 0}^n a_k^2 ?$$ For example, $\sum_{k = ...
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82 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
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89 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
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33 views

Lattice of ideals in a number field

Let $n = [K : Q]$ be the degree of a number filed $K$ and let $(r_1, r_2)$ be the signature of $K$. Let $σ_1, . . . , σ_{r_1}$ be the $r_1$ real embeddings of $K$ into $\mathbb R$. We choose one of ...
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124 views

Divisor summatory function for squares plus one

As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that: $\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$, where $\tau(n)=\sum_{d|n}1$ ...
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75 views

Question about application of Erdős-Kac theorem

My question is whether (*) below can be shown using the Erdős-Kac theorem? I don't think the distinction between $\Omega$ and $\omega$ is important here. For lack of better notation let ...
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96 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
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Is this “by symmetry” statement valid?

Problem: Let $p,q,r$ be integers such that $\gcd(p,q,r)=1$. Prove that there exists an integer $A$ such that $\gcd(p,q+Ar)=1$. A start: Assume for the sake of contradiction that $\gcd(p,q+Ar)>1$ ...
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43 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ...
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Reference request for Fourier analysis on local fields

I am studing Class field theory. I need a good reference books, notes e.t.c which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier ...
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57 views

What is known about$\sum\limits_{p\text{ prime}} \frac{1}{p^2-1}$?

Are there some known results for $\sum\limits_{p\text{ prime}} \dfrac{1}{p^2-1}$? I wasn't able to find anything about this sum in the internet or in my books!
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29 views

How many zeros within a number

Let noughts(n) be the number of noughts needed to write n in base 10.If n is given how can I find out the value of noughts(n) . I myself have tried to compute noughts(n) by examining all the digits ...
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113 views

Pell's Equation and the Pigeon Hole Principle

David Speyer gave a beautiful application of the pigeon hole principle here to show that Pell's equation $$x^2-Dy^2=1$$ has infinitely many integral solutions. I was wondering if anybody knows the ...
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58 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
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60 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
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How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
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77 views

Sum of roots (number theory)

Let k,m∈ℕ. Let a1,a2,...,ak>0 and b1,b2,...,bm>0. Let for all natural n, n>1 Prove that k=m. Prove that a1a2...ak=b1b2...bk Prove that if each of the two sets of numbers sort of growth, then ...
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45 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
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97 views

A question on the Prime number theorem

Let $N\geq1$. Could we infer $$\sum_{n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$$from $$\sum_{n\leq N}\Lambda(n)= N+O(N\exp( -c\sqrt{\log N})$$or $$\sum_{p \leq N}1=Li(x)$$ without resorting to the ...
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86 views

how to find the last non-zero digit of $n$

I want to know how to find the last non-zero digit of $n$. For example $n = 100!$ my try: First i have to know how much Zeros $100!$ has so i did this: $$E_{5}100 = \sum _{1\leq k <n} ...
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63 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
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46 views

Is every even integer $\geq 6$ the sum of two totients?

1) Is it true for every even integer $n \geq 6$ that there exist integers $x, y$ such that $$n = \varphi(x) + \varphi(y)?$$ For example: $6 = 2+4 = \varphi(3) + \varphi(5),$ $8 = 4+4 = \varphi(5) + ...
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59 views

Bessel function sum?

Is $$f(x)=\dfrac{2\pi}{n^2}\sum_{m=1}^{x}\sum_{k=1}^{n}\dfrac{\Im(e^{\dfrac{\pi i m}{\Im(\rho_k)}})k}{\Im(\rho_k)}$$ related to the Bessel family of functions? Plot for $n=300$ Or is it related ...
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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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43 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
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72 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
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49 views

How can I compute |R/I| where $R\simeq\mathbb Z^n$ and $I\unlhd R$?

A number field $K$ is a field $\mathbb Q\le K\le\mathbb C$ of finite degree over $\mathbb Q$, say $n$. Call $\mathbb A$ the ring of algebraic integers of $\mathbb C$; an algebraic integer is an ...
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how that if $P(x_1,…,x_n) \in C[x_1,…,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant.

Show that if $P(x_1,...,x_n) \in C[x_1,...,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant. To start, how would you express $P(x_1,...,x_n)$? I really ...
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80 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 4

Following a previous question (here you'll find an introduction): A paper by Maier which refutes Cramer's Model suggests we should replace the heuristic "$\Bbb P(n\in\mathcal P)=1/\log n$" with ...
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68 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
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80 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
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71 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
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75 views

Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
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194 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...