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

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2
votes
2answers
76 views

Which are the two last digits of $a_n$?

We have the sequence $(a_n)$ with $a_1=3$ and the recursive formula: $$a_{n+1}=3^{a_n} , \forall n$$ Which are the two last digits of $a_n$ ? How can I find them? I have to find $a_n \pmod {100}$ ...
0
votes
2answers
38 views

Dirichlet Characters and Chineese remainder theorem

Let $k=k_1 k_2$ s.t. $(k_1,k_2)=1$ and let $\chi$ be a dirichlet character mod $k$. I'm trying to prove that there exsists $\chi_1,\chi_2$ dirichlet characters mod $k_1,k_2$ respectively, s.t. ...
1
vote
0answers
31 views

Ramanujan Class Invariant $G_{125}$ and $ G_{5}$

How to calculate the Ramanujan Class Invariant $G_{125}$ and $G_{5}$?
2
votes
1answer
53 views

System,is my solution right?

Solve the system: That's what I have tried: The system is equivalent to this one: $$\text{We set: } M=5 \cdot 2 \cdot 7 \cdot 11,M_1=154,M_2=385,M_3=110,M_4=70$$ We solve the following: $$ ...
6
votes
3answers
202 views

Number Theory Reading List

What are the essential number theory texts that every serious student of number theory should read?
1
vote
1answer
64 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
0
votes
1answer
28 views

About the congruence.

I dont know how to prove this question. I can only get $a^{m^s} ≡ b^{m^s}$ ( mod $m^r$). Suppose $m$ and $r$ are positive integers. If $a ≡ b$ (mod $m^r$ ), then for each $s \geq 0$, $a^{m^s} ...
12
votes
1answer
151 views

Does this function change signs infinitely often?

$$f(n) = \sum_{i = 1}^n (-1)^{\omega(i)}$$ where $\omega(n)$ counts how many distinct prime factors $n$ has. I don't see any sign changes past $n = 49$, but I've only computed it up to $n = 1,000$.
0
votes
1answer
39 views

Minimum AND operation on subset

Given an array of size N . Let's create all the subsets of this array which contain at least 2 elements. Now, operate AND over the elements of each subset, and store the results in a new array. I ...
3
votes
1answer
54 views

$a^p \equiv b^p \pmod p \Rightarrow a^p \equiv b^p \pmod {p^2}$

I am looking at the following exercise and I got stuck.. If $p$ is prime, $p \nmid a$, $p \nmid b$, prove that $$a^p \equiv b^p \pmod p \Rightarrow a^p \equiv b^p \pmod {p^2}$$ Could you give me ...
5
votes
3answers
63 views

$a^{12} \equiv 1 \pmod{35}$,knowing that $(a,35)=1$

Prove that $\forall a \text{ with } (a,35)=1:$ $$a^{12} \equiv 1 \pmod{35}$$ $$35 \mid a^{12}-1 \Leftrightarrow 5 \cdot 7 \mid a^{12}-1 \overset{(5,7=1)}{ \Leftrightarrow} 5 \mid a^{12}-1 \text{ and ...
2
votes
0answers
59 views

An unsolvable Number Theory Question [duplicate]

This is a question I was sent by a friend, I really have zero idea on how to solve the question. I'm not even sure where to begin solving it. Though I did make an attempt to solve it, I always reached ...
1
vote
1answer
47 views

$\binom{2n}{n} \equiv 0 \pmod{p}$

Let a prime $p$ such that $1 \leq n <p<2n$. Prove that: $$\binom{2n}{n} \equiv 0 \pmod{p}$$ Can I do it like that? $$\binom{2n}{n}=\frac{(2n)!}{n!n!}=\frac{(n+1)(n+2) \cdots (2n-1)2n}{1 \cdot ...
1
vote
1answer
30 views

Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
1
vote
1answer
38 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
5
votes
1answer
177 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
4
votes
1answer
208 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
1
vote
1answer
28 views

Can you derive a formula for the semiprime counting function from the prime number theorem?

E.g., their are $4$ semiprimes less than or equal to $10$ $(4, 6, 9, 10)$ or $2$ squarefree semiprimes ($6$ and $10$). It's ok if it's off for small numbers but gets more accurate as $n \to \infty$.
1
vote
3answers
63 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
4
votes
5answers
93 views

How can I show that $\phi(m) \mid \phi(n)$? [duplicate]

I want to prove that: $$\text{ if } m,n \geq 1 \text{ and } m \mid n,\text{ then } \phi(m) \mid \phi(n).$$ How can I show this? I thought the following: $$m \mid n \Rightarrow \exists k \in ...
0
votes
2answers
40 views

The product of two numbers that can be written as the sum of two squares

Prove that if n is the product of two numbers that can be written as the sum of two squares then n can be written as the sum of two squares.
0
votes
1answer
43 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
-2
votes
1answer
90 views

To solve $x^2+y^2=(x-y)^3$ in integers [closed]

How do we find all $x,y \in \mathbb Z$ such that $x^2+y^2=(x-y)^3$ ?
1
vote
1answer
39 views

Positive integers $n$ satisfying $\omega(n)<\omega(n+1)<\omega(n+2)$

Let $ \omega(n) $ denote the no. of distinct prime divisors of $n$ , then does there exist infinitely many positive integers $n$ satisfying $\omega(n)<\omega(n+1)<\omega(n+2)$ ?
1
vote
3answers
39 views

To prove $a^2+ab+b^2 \Big|(a+b)^{2n}+a^{2n}+b^{2n}$ whenever $3$ does not divide $n$

If $3$ does not divide a positive integer $n$ , then how to prove that $a^2+ab+b^2 \Big|(a+b)^{2n}+a^{2n}+b^{2n}$ ?
5
votes
3answers
76 views

$4^x+6^x=9^x$ $\implies$ $x \notin \mathbb Q$?

Does there exist any rational number $x$ such that $4^x+6^x=9^x$ ?
0
votes
2answers
35 views

Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.

Consider numbers of the type $n=2^m+1$. Prove that such an $n$ is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime. Consider numbers of the type $n=a^m-1$ where $a>1$ is a ...
1
vote
1answer
19 views

Show that $1*1=\tau$, $I*I=I \times \tau$, $1*I=\sigma$, $\mu * \tau=1$, and $\mu * \sigma=I$

Show that $1*1=\tau$, $I*I=I \times \tau$, $1*I=\sigma$, $\mu * \tau=1$, and $\mu * \sigma=I$. I am not familiar with the use of these symbols. This functions are all multiplicative arithmetic ...
2
votes
2answers
45 views

Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$

Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$ and conclude that $\phi(n)=n \sum_{d|n} \frac{\mu(d)}{d}$. Proof of $\sum_{d|n} ...
3
votes
1answer
44 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} ...
2
votes
0answers
205 views

Count swap permutations

Given an array A = [1, 2, 3, ..., n]: ...
0
votes
0answers
18 views

Suppose f is an additive real valued arithmetic function, and let $a>0$ be a real number.

Suppose f is an additive real valued arithmetic function, and let $a>0$ be a real number. Define F by $F(n)=a^{f(n)}$ for all $n \epsilon N$. Prove that F is a multiplicative arithmetic function. ...
1
vote
0answers
91 views

Curious set of $n\sin^2(n)$

I consider this set $S_{0}=\{ n \in \mathbb{N}: n\sin^2(n) < 1 \}$. And I have some questions. See the elements of $S_{0}$= $\{ 1,3,6,19,22,25,44,47,66,69,88,110,132,154,157,176,179,$ (common ...
2
votes
0answers
18 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
1
vote
2answers
60 views

Prove that diophantine equation has only two solutions.

I am looking at the following exercise: $$\text{Prove that the diophantine equation } x^4-2y^2=1 \text{ has only two solutions.}$$ That's what I thought: We could set $x^2=k$,then we would have ...
0
votes
3answers
52 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
-1
votes
2answers
44 views

$a,b \in \mathbb N $ , $b$ odd $\implies$ $ \dfrac{2a^2-1}{b^2+2} \notin \mathbb Z $?

If $a,b$ are positive integers and $b$ is odd , then is it ever possible that $ \dfrac{2a^2-1}{b^2+2} $ is an integer ?
3
votes
1answer
77 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
-1
votes
1answer
29 views

Show that $n=\gamma(n)$ if and only if $\mu^2(n)=1$. [closed]

How can I show that $n=\gamma(n)$ if and only if $\mu^2(n)=1$ where $\gamma(n)=\prod_{p|n}$ and $\mu$ is möbius function ? I know that $\gamma$ and $\mu$ are both multiplicative functions for more ...
1
vote
1answer
97 views

Can any real number be expressed as an integral combination of $e$ and $\pi$?

Prove or disprove: For any real number $x$, there exist integers $a$ and $b$ such that $ae + b\pi=x$. It certainly seems improbable, but how does one prove it?
1
vote
1answer
67 views

Prove that for any multiplicative arithmetic function $\phi$ we have $\phi(1) = 1$.

Prove that for any multiplicative arithmetic function $\phi$ we have $\phi(1) = 1$. I know there is a clever way to prove this by it is not coming to mind. Definition: An arithmetic function $f : ...
0
votes
2answers
32 views

$\log$ approximation for $\pi(x)$

It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where $f(y, x) := \log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor ...
1
vote
1answer
53 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
vote
2answers
53 views

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
0
votes
3answers
58 views

Disproof of Gelfond-Schnieder Theorem [closed]

The Gelfond Schneider theorem somewhere says that "There exist 2 such irrational numbers a and b(where a doesn't equal to b), ab is rational. The solution is taken as (in many answers in stack ...
2
votes
1answer
72 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
2
votes
1answer
106 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 ...
3
votes
2answers
37 views

Continued fraction and classification of real numbers.

I would be grateful if anyone can tell if there are any methods to classify real numbers using continued fraction. eg: Suppose $[a_0;a_1,a_2,\ldots,a_n]$ is the representation of some real number ...
4
votes
4answers
218 views

How to show a group is cyclic?

One question asking if $\mathbb{Z}^*_{21}$ is cyclic. I know that the cyclic group must have a generator which can generate all of the elements within the group. But does this kind of question ...
1
vote
0answers
31 views

On non-divising primes of an integer $x$

We know more about divisor than non-divisors, If we consider the sets : $$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set) $$ P^{2}_{x} =\left \{ ...