Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

learn more… | top users | synonyms (1)

4
votes
0answers
109 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 ...
4
votes
0answers
514 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 ...
4
votes
0answers
99 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 ...
4
votes
0answers
132 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 = ...
4
votes
0answers
337 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. ...
4
votes
0answers
699 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 ...
4
votes
0answers
148 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 ...
4
votes
0answers
74 views

Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$

I'm looking for composite $n$ such that $$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$ Are there only finitely many? Can this be proved? This is Sloane's A070037 but there's not much information in the ...
4
votes
0answers
304 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?
4
votes
0answers
130 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 ...
4
votes
0answers
131 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 ...
4
votes
0answers
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 $p=3$, ...
4
votes
0answers
369 views

Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
4
votes
0answers
693 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 ...
4
votes
0answers
181 views

Is this a recurrence for the Mertens function plus 2?

If we define a symmetric array: $$T(1,1)=3,\; T(1,2)=2,\; T(2,1)=2$$ $$T(1,k)=\frac{-T(n,k-1)-\sum\limits_{i=2}^{k-1}T(i,k)}{k+1}+T(n,k-1)$$ $$ ...
4
votes
0answers
130 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} ...
4
votes
0answers
278 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: ...
4
votes
0answers
383 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 ...
4
votes
0answers
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 ...
3
votes
0answers
51 views

Are there infinitely many $N^3$ (especially for prime $N$) that cannot be expressed as a sum of three positive cubes?"

Well few days ago i asked a question on perfect numbers and Tito Piezas III answered the question in a very intriguing way which has helped me to get a lead on it.But his answer and perfect numbers ...
3
votes
0answers
59 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: ...
3
votes
0answers
107 views

What is the value of $\sum_{p\le x} 1/p^2$?

My question is, what is the value of $$\sum_{p\le x} \frac{1}{p^2}?$$ More generally, what is the value of $$\sum_{p\le x} \frac{1}{p^n}?$$ How can we find it? For $\sum_{p\le x} 1/p$ the idea was ...
3
votes
0answers
116 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)\tag1$$ Finding certain equal sums of like powers that are multi-grades for high powers lead to two questions: If ...
3
votes
0answers
41 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 ...
3
votes
0answers
30 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 ...
3
votes
0answers
76 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 ${\tau}^n = id$ ?
3
votes
0answers
58 views

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 = ...
3
votes
0answers
25 views

$GL_2(\mathbb{Q}_p)$ and $GL_2(\mathbb{Z}_p)\cdot\mathbb{Q}_p^\times$

I am confused by a question, which is probably of school level. In some papers I have seen an induction from the group $GL_2(\mathbb{Z}_p)\cdot\mathbb{Q}_p^\times$ to the group $GL_2(\mathbb{Q}_p)$, ...
3
votes
0answers
84 views

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 $n\ge 3$, ...
3
votes
0answers
79 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 ...
3
votes
0answers
67 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 + ...
3
votes
0answers
26 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 ...
3
votes
0answers
103 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 $\tau(n)=\sum_{d|n}1$ ...
3
votes
0answers
73 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 ...
3
votes
0answers
67 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 ...
3
votes
0answers
29 views

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 $\gcd(p,q+Ar)>1$ ...
3
votes
0answers
37 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 ...
3
votes
0answers
48 views

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 ...
3
votes
0answers
28 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 ...
3
votes
0answers
54 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 & & ...
3
votes
0answers
51 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 ...
3
votes
0answers
74 views

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) = ...
3
votes
0answers
76 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 ...
3
votes
0answers
40 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{?}= ...
3
votes
0answers
88 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 ...
3
votes
0answers
56 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} ...
3
votes
0answers
49 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 ...
3
votes
0answers
53 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 ...
3
votes
0answers
41 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 ...
3
votes
0answers
57 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. ...