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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Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
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263 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 $n-1=2^xc$ ...
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73 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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104 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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113 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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149 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 ...
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357 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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408 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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49 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 ...
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120 views

Solutions to simultaneous excluded congruences

I'm interested in the smallest solution to a family of "excluded congruences." To be precise, let $p_1 < \ldots< p_k$ be a sequence of primes and consider the constraints $$ x \not\equiv a_1 ...
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343 views

Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
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162 views

Does Hensel prove LTE?

Can the Lifting The Exponent Lemma ( found at http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf) be proven with Hensel's Lemma? How?
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283 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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117 views

Number Theoretic Game

2 players A and B play a game. At the start of the game, $n$ positive integers (not necessarily distinct) are written on a notebook. First, player A chooses a number from the notebook and declares it ...
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520 views

Obtain a contradiction

Motivation : The motivation is to show that the equation $x^{2b}.x^{2a} +(3-x^{2b}) x^{a} + (1-s^2)=0 $ has no solutions in integers for any values of $x,b,a,s$ ( choosen as per the constraints ...
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102 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
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135 views

Liftings in unramified extensions of $\Bbb Z_p$

[Edit : I have changed the formulation of the question. Sorry for the trouble] Here is a stupid question, maybe trivial. Let $p$ be a prime number, $q = p^n$ where $n$ is an integer, $R = ...
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367 views

sum of $n^{th}$ powers of prime factors of $x$

Starting with a positive integer $x$, find the sum of the $n^{th}$ powers of the prime factors of $x$, including multiplicities. Then find the sum of the $n^{th}$ prime factors of the result etc. ...
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725 views

Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
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149 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
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312 views

Sum of the primitive roots

It is well known that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \bmod{p}$. But I can't see why this result is interesting or useful. Can someone please enlighten me?
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135 views

How to prove that polynomials with integer coefficients generically have full Galois group

Based on the graphic in the MathWorld article on the quintic equation it seems very likely that following statement is either true or trivially false in a way that can be easily remedied by adding an ...
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147 views

Champernowne-like squares, are there any?

I read about the Champernowne constant on Wikipedia a couple of days ago, and I got curious about something similar: is there some "Champernowne-like" number; that is, a concatenation of all numbers ...
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138 views

Eigenvalues of the $p$-adic Harmonic oscillator?

Given a prime $q$, what are the values of the $p$-adic Harmonic oscillator that is the solution to the following $p$-adic differential equation? $$ -D^2_q f(x)+ x_q^2 f(x) = E_n f(x) .$$ What are ...
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47 views

When does PSL(5,q) have order exactly divisible by a specific odd power of 5?

In a misguided attempt at answering a question on divisibility of simple group orders, I looked at $\newcommand{\PSL}{\operatorname{PSL}}v_p(|\PSL(p,q)|)$ which went smooth enough for $p=2$ and $p=3$, ...
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63 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?
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703 views

Project Euler Problem 338

I'm stuck on Project Euler problem 338. This is a cross post from StackOverflow where I initially posted, however, it was suggested that I post it here too since the problem mostly relies on math. The ...
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182 views

Is this a recurrence for the Mertens function plus 2?

If we define a symmetric array: $$T(1,1)=3,\; T(1,2)=2,\; T(2,1)=2$$ $$T(1,k)=\frac{-T(n,k-1)-\sum\limits_{i=2}^{k-1}T(i,k)}{k+1}+T(n,k-1)$$ $$ ...
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135 views

Hecke operators as endomorphism of Jacobians of modular curves

Let $p$ be a primes that does not divide $N$, then $T_p$ defined an endomorphism $J_0(N)\to J_0(N)$. what is $T_p^\vee$? In other words, we naturally have $J_0(N)^\vee \xrightarrow{T_p^\vee} ...
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282 views

Prime zeta definition, multiplication by zero

Wikipedia has a page about the prime zeta function which is defined as follows: $$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$ I entered this additional definition: Define a sequence: ...
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401 views

p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a ...
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92 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
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39 views

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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26 views

Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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33 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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42 views

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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63 views

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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36 views

A question on the Pell equation $x^2-5dy^2 = -1$

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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34 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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21 views

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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217 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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40 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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41 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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38 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
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66 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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48 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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40 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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25 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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90 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 ...