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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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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373 views

Can $\sigma_4(n)$ be a square number?

Denote $\sigma_k(n)=\sum_{d\mid n}d^k,$ then $$\sigma_1(3)=2^2,\sigma_2(42)=50^2,\sigma_3(2)=3^2,$$ and both $\sigma_1(345)$ and $\sigma_3(345)$ are square numbers: ...
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97 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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122 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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170 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
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89 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
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121 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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131 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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212 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
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406 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 ...
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86 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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87 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 = ...
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85 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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190 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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172 views

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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264 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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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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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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152 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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361 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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436 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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121 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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372 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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292 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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120 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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523 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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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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406 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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740 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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316 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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137 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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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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708 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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183 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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283 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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409 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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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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29 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
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37 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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65 views

Zariski density of points over completion

I have a simple question which I couldn't find a reference to. Let $X$ be a smooth projective irreducible variety over $\mathbb{Q}$. Suppose we base change to $\mathbb{Q}_p$ (the $p$-adics) and ...
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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 ...