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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Solving exercise 1.10 in Silverman's AEC

Please note that although there is a very similarly titled question Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves" this question received no answers. Let $p$ be an odd prime and ...
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Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
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Town Network - Show there is a Tetrahedron of the Same Transport

Question: There are 18 towns such that between each pair of towns there is either a train or bus service (not both). Prove that there are 4 towns such that all 6 of their pairwise connections use the ...
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Are there any known special properties of a number located between twin primes?

With the exception of $4$, every number located between twin primes is divisible by $6$. This one is obvious, but are there any other properties that can be ascribed to such numbers? A property may ...
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degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
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$ \lim_{n \to \infty} \sum_{i=1}^n\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( log(2 \pi [n/i])){\mu{(i)}} = \beta $

I'm just a physics undergraduate. I think (in an non-rigorous way) I have managed to prove: $$ \lim_{n \to \infty} \sum_{i=1}^n\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( ...
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On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
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How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
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A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
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35 views

Merten's function

I am tasked with applying the Wiener-Ikehara Theorem to achieve a bound of little o(x) on Merten's function $\sum_{n=1}^x \mu (n)$. My problem is the Wiener-Ikehara Theorem applies to Dirichlet series ...
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Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
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Can we treat finite and infinite primes in scheme theory?

In Number Theory, I know we have analogy to geometry by considering infinite primes not only Spec $\mathcal{O}_K$ (and I heard the point of view bring the Riemann-Roch theory). My question is can we ...
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219 views

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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46 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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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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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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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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<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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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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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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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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151 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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62 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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66 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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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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Comparing up-arrow's

Is it true that $$3\uparrow^{n+1} 3\ >\ n\uparrow^n n $$ holds for every $n\ge 1$ Since $3\uparrow^{n+1}3=3\uparrow ^n 3\uparrow ^n 3$ and $3\uparrow^n3$ is much bigger than $n$ for $n\ge 3$, ...
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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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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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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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113 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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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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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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40 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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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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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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101 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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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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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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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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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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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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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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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 ...