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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1answer
63 views

Show that the equality is true

If $f$ is a Completely multiplicative function and $g$ is an arithmetic function such as $g(1) \neq 0$ prove that: $$(f\cdot g)^{-1} = f\cdot g^{-1}$$ Any function with the -1 as exponent is the ...
2
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1answer
42 views

How to show this equality

If $f$ is a multiplicative function and ¨$n$¨ is a square-free positive integer. Prove that: $$f^{-1}(n) = \lambda(n)\cdot f(n)$$ where $f^{-1}$ is the dirichlet inverse and $\lambda$ is the ...
2
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1answer
39 views

Meaning of congruence notation for Bernoulli Numbers

I am studying Theorem 4(von Staudt's Theorem) in Borevich-Shafarevich's Number Theory(1966)(page 384) which states: Let $p$ be a prime and $m$ an even integer. If $(p-1)\nmid m$, then $B_m$ is $...
2
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1answer
56 views

Show that the first derivative of the Riemann Zeta function $\zeta'(s) < 0$ if $s \in (1-\epsilon,1)$ and $\epsilon > 0$ is sufficiently small.

Show that $\zeta'(s) < 0$ if $s \in (1-\epsilon,1)$ and $\epsilon > 0$ is sufficiently small. Using the fact that \begin{align} \zeta(s) = \frac{s}{s-1}-s\int_1^\infty\frac{\{t\}}{t^{s+1}}dt \...
2
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1answer
118 views

Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
2
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1answer
56 views

The number $2^p-2$

Suppose $p$ is a prime number. Use computer, for all $p<10^7$, I find that almost all the valuation of $2^p-2$ at $p$ is one, only two exceptions: $p=1093$ or $3511$, at these two primes the ...
2
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1answer
84 views

definition of half-integral weight modular forms

i start reading about modular forms of half-integral weight $k/2$ for $\Gamma' \subset \Gamma_0(4)$. As far as i understand these are holomorphic functions $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
2
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1answer
178 views

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
2
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1answer
119 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
2
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3answers
118 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
2
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1answer
84 views

Is every positive integer the sum of at most 8 pentatope numbers?

Is every positive integer the sum of at most 8 pentatope numbers ? See : http://en.wikipedia.org/wiki/Pentatope_number I saw this conjecture here : http://math.eretrandre.org/tetrationforum/...
2
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1answer
46 views

Finding the lowest number (or an upper bound to the lowest number) not congruent to a set of moduli

Note: if finding x is not possible, an upper bound, where there must be at least one number less than said number which is not congruent to the set, would be helpful. The set: For my purposes, the ...
2
votes
1answer
50 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
2
votes
3answers
301 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of $M_n$,...
2
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1answer
87 views

Generalization for Stirling numbers 2nd kind to negative column-indexes?

The exponential generating functions for the Stirling numbers 2nd kind are the n'th powers of $f(x)=\exp(x)-1$ (where this is understood as formal power series, Abramowitz&Stegun, 26.8.12). ...
2
votes
1answer
88 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 \...
2
votes
3answers
70 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
2
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1answer
58 views

Evaluate the Legendre symbols $(\frac{503}{773})$ and $(\frac{501}{773})$

Evaluate the Legendre symbols $(\frac{503}{773})$ and $(\frac{501}{773})$. My solution: $(\frac{501}{773}) = (\frac{167 \cdot 3}{773}) = (\frac{167}{773}) \cdot (\frac{3}{773}) = (\frac{773}{167}) \...
2
votes
1answer
91 views

Sums involving square of Moebius function

I try to estimate the following sum: $$ \sum_{n \leq x}\mu(n)^2 f(n) $$ where $\mu(n)$ is a Moebius function and $f(n)$ is some multiplicative arithmetic function. If I understand it correctly it is ...
2
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1answer
101 views

A formula occurring in Dirichlet's proof of the infinity of primes in an AP.

While studying Dirichlet's proof of an infinity of primes in any AP with first term and common difference coprime, the formula below involving the gamma function was quoted as being well known. $$\...
2
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2answers
26 views

How can I show that ord_n(a^u) is equal to t/(t,u)

I'm having a hard time on this question: Let n > 0, a $\neq$ 0, and (a,n) = 1. If ord$_n$ a = t, and if $u$ is a positive integer, then show that: $$ord_{n}(a^{u}) =\frac{t}{(t,u)}.$$ So far I've ...
2
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1answer
398 views

Understanding a proof of Lagrange's four-square theorem

I've been looking at Wikipedia's proof of the four-square theorem and trying to work out the details - I like that it doesn't need to separate the cases for $m$ even and odd, but there is one step ...
2
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1answer
77 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
2
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1answer
266 views

Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
2
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1answer
55 views

How to get the numbers that multiplied with $N$ become integers?

How can I get all the numbers (between $0$ and $1$) that multiplied with integer $N$ become integers. Thanks!
2
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1answer
64 views

Geometric generalizations of the arithmetic and geometric means

Given a set of $3$ positive real numbers $\{a, b, c \}$, one way of interpreting the geometric mean is that it is the answer to the following question: Find the edge length $\mu_3$ of a cube whose ...
2
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1answer
141 views

Prove that $\sigma_k$ is a multiplicative function

For each real $k$,we define: $$\sigma_k(n)=\sum_{d \mid n} d^k$$ $$\text{Prove that } \sigma_k \text{ is a multiplicative function.}$$ That's what I have tried: $$\sigma_k(1)=\sum_{d \mid 1} d^k=...
2
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1answer
92 views

Finding Prime triples with $p_{n} +p_{n+1} −p_{n+2} = 1$

I was just looking at a sequence of primes and suddenly I got this thought that $p_2 +p_3 −p_4 = 1$ since $p_2 = 3, p_3 = 5, p_4 = 7$. Also for $p_3 = 5, p_4 = 7, p_5 = 11$ one has $p_3 +p_4 −p_5 = 1$....
2
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1answer
119 views

A term of the sequence

Let $a_0$ and $a_n$ be diff erent divisors of a natural number $m$, and $a_0, a_1, a_2,\cdots, a_n$ be a sequence of natural numbers such that it satisfies $$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 ...
2
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1answer
54 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
2
votes
1answer
51 views

Based on prime number theorem

I have a problem... If $A(x)=\sum_{n\leq x}a(n)$ where $a(n)=\frac{1}{k}$ if $n=p^{k}$ and $a(n)=0$ in other case, show that $A(x)=\pi(x)+O(\sqrt{x}\:log\:logx)$ I think I should use the theorem of ...
2
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1answer
72 views

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$?

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$ ? And why does $p_i^{\alpha_i} \mid \text{ord}_n(a)$ ? I see that $\text{ord}_n(a)$ ...
2
votes
1answer
58 views

On composite-ness of numbers of certain form

If $a\ge750$ is an integer , then is it true that $\bigl(2\times17^{4a+3}\bigr)+1$ cannot be a prime ?
2
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1answer
74 views

Prime or composite?

Factorials are very interesting to solve. How will you find that $2014!+1$ (where '$!$' means factorial) is prime or composite ?
2
votes
2answers
152 views

Diophantine equation with cubes.

Interested in the solution in general Diophantine equations of the form: $X^3+Y^3+Z^3=3XYZ+q$ $q$ - what some integer. Solutions similar equations can be written. Since this equation is easy, as ...
2
votes
1answer
62 views

What is the order of the product of $ \frac{p-1}{p} $ under the square root of a prime?

Is there any known asymptotical order for $$ \prod_{p_k\ \text{prime}}^{\sqrt{p_n}} \frac{p_k-1}{p_k} $$
2
votes
1answer
60 views

Techniques in solving the congruence modulo

How can the congruence ${x}^{17389}\equiv43927 \pmod{64349}$ be solved? I read that the first step is to solve the congruence $17389d\equiv1 \pmod{63840}$. I think $d$ is a number such that $17289d≡1 ...
2
votes
1answer
349 views

Algorithm for checking Prime Power

Suppose we are given some arbitrary positive integer. How can we check whether the integer is a prime power? Brute force would be very inefficient in this case.
2
votes
1answer
204 views

How to show that the $x^a \equiv 1 \pmod p$ has exactly $\gcd(a,p-1)$ solutions at $Z^*_{p}$?

It is given that $p$ is prime number and $a\ge1$ solution so far: $x^{\gcd(a,p-1)} ≡ 1$ because it known that a group of units of $Z/pZ$ is cyclic and of order $n=p-1$ for $p$ prime, and also in ...
2
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1answer
68 views

$(x^{2014}-x^{2004})\in \mathbb Z , \;(x^{2009}-x^{2004})\in \mathbb Z$ $\implies$ $x \in \mathbb Z$?

Let $x$ be a real number such that $(x^{2014}-x^{2004})$ and $(x^{2009}-x^{2004})$ are both integers. Then is $x$ also an integer ?
2
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1answer
89 views

rational parameterization of quartic

With the curve $x^4 - 6x^2 - y^2 + 1 = 0$ in the range of $x$ inside of $(-1,1)$, I can only identify two rational points $(0,1)$ and $(0,-1)$. Is it possible to determine if there are others?
2
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1answer
28 views

The distribution on the number line of the sum of squares

If $x,y,z$ are all non-negative integers, then what is the density of $x^2+y^2+z^2$ on the number line near $N$? From the plot it seems pretty much linear, i.e. $dn/dN=C$ where $C$ is some constant ...
2
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1answer
48 views

In how many ways can a number be factorized over the field $\mathbb{Z}_p$ into two numbers?

For example, over the field $\mathbb{Z}_5$, we can factor number 4 into two numbers in three different ways, i.e. 4=4$\times$1, 4=2$\times$2, and 4=3$\times$3. I am looking for a general formula of ...
2
votes
1answer
37 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4\cdot \log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : $$\theta(n) \le \theta(2^{k+1}) < 4\cdot \log[2n]$$ by definition we have $$\...
2
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1answer
31 views

How can I conclude that the integers $a r_1, a r_2 … ar_{\phi(m)}$ modulo $m$ are a permutation of the integers $r_1,r_2,…,r_{\phi(m)}$?

How can I conclude that the integers $a r_1, a r_2 ... ar_{\phi(m)}$ modulo $m$ are a permutation of the integers $r_1,r_2,...,r_{\phi(m)}$ given the proofs that $(ar_i,m) = 1$ ($\gcd$) for every $i$ ...
2
votes
1answer
111 views

finding square $k$ such that $k\mid 2^{k-1} -1$

Can we find a positive square integer $k>1$, which satisfies $k\mid 2^{k-1} -1$ ? If yes, what are such $k>1$ values? Here $k = n^2$ and $n$ is some positive integer. If we cannot find such ...
2
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1answer
32 views

Properties of divisibility

I would like to know if it's true if $a|b$ and $c|d$, then $ac|bd$. I prove in this way: if $a|b$ and $c|d$, then there are $k_1$ and $k_2$, such that $b=k_1a$ and $d=k_2c$, thus $bd=k_1k_2ac$ and $ac|...
2
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1answer
201 views

Is my conjecture correct? Any advice on how to solve this conjecture?

I was doing problem 6.3 from here. To make this less programming and more math oriented: GCDMany is equivalent to using Euclid's method (using mods and NOT ...
2
votes
1answer
57 views

$ax^2 + b$ and infinitely many primes: Does existence proof exist?

The question is on a subset of Bunyakovsky's Conjecture on an infinite number of primes existing in integer polynomials of degree higher than $1$. The conjecture itself is open. I have not been able ...
2
votes
1answer
69 views

Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer sufficiently large N such that ...