4
votes
0answers
24 views

Arithmetic Functions: Evaluate $ \sigma(210)$ and $d(63)$

Evaluate $ \sigma(210)$ and $d(63)$ I'm not sure if I got this correct, so here is my attempt. By Theorem 6.3, suppose we have $n=p_1^{\alpha 1}...p_s^{\alpha s}$, then $d(n) =(\alpha_1 ...
3
votes
1answer
32 views

Eulers totient function divided by $n$, counting numbers in the set [1,m] that are coprime to n

If we divide Euler's totient function $\omega(n)$ by $n$, we obtain a fraction. If we multiply this fraction by any natural number $m$ which gives us another natural number $p$, is it true that $p$ is ...
1
vote
0answers
29 views

Show that $-1$ is a square $\mod n$, if $n\equiv 1\mod 4$?

I am trying to prove that $-1$ is a square modulo $n$ if, and only if $n\equiv 1\mod 4$. One direction i think i have done... So, we have that $n\equiv 1\mod 4$, from this follows that $n$ must be ...
2
votes
0answers
25 views

A primitive root exists modulo $n$ if and only if $n=2$, $n=4$, $n=p^k$, or $n=2p^k$ with $p$ an odd prime.

I have already proven that primitive roots exist modulo $p^k$ and $2p^k$ for an odd prime $p$. I'm having trouble proving the other direction. Is it simply due to the fact that if $p,q$ are distinct ...
0
votes
0answers
12 views

Partial sums of powers of the divisor function

It is easy to establish that $$\sum_{n\le x}\tau(n) \sim n\log n$$ How would one find good bounds on $$\sum_{n\le x} \tau(n)^k $$ for some $k > 0$
2
votes
1answer
44 views

Fermat solved $x^2+2=y^3$ by infinite descent?

In a letter to Christiaan Huygens entitled "on problems in the theory of numbers: a letter to Christiaan Huygens", Fermat claism that he solved the diophantine $x^2+2=y^3$ using infinite descent. Here ...
2
votes
1answer
29 views

Definition of field of fractions of $p$-adic integers

I am given this definition of the field of fractions of the p-adic integers: $$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$ How can I show that: $Q_p$ consists of the sums of the ...
0
votes
0answers
19 views

Calculate sum of distinct pairs [on hold]

Given an array A we need to find the sum of all distinct pairs of indexes from the array and adds the value ⌊$A[i]+A[j]\over A[i]×A[j]$⌋ to the sum Note: ⌊$A\over B$⌋ is the integer division ...
3
votes
4answers
100 views

Prove that $2^{10}+5^{12}$ is composite

Prove that $2^{10}+5^{12}$ is composite I need to solve this using only high school mathematics. Any ideas?
2
votes
1answer
37 views

Number of integers of the form $3k+1$ in range $[a,b]$ [on hold]

How do I find the number of integers in the range $[a, b]$ that are of the form $3k+1$, where: $a,b,k$ are natural numbers. $a \le b$
2
votes
3answers
58 views

$\dfrac1a+\dfrac1b=\dfrac1c$, $a, b, c \in \mathbb{N}$ with no common factor, find all solutions [duplicate]

Given $\dfrac1a+\dfrac1b=\dfrac1c$, where $a, b, c \in \mathbb{N}$ with no common factor, find all solutions. Actually, you can think this question as a follow up of this one. Today, I saw this ...
0
votes
2answers
25 views

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$? I know this question is quite trivial and I will understand if it gets removed. I am trying to ...
0
votes
1answer
31 views

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent??

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent? So far I have that \begin{align} \sum\limits_{\text{p prime}, p \geq 2} ...
2
votes
1answer
33 views

The congruence has a solution

Sentence: If $a \in \mathbb{Z}$, then the congruence $x^2=a \pmod p, \forall p \in P$ has a solution $\Leftrightarrow$ $a=\square$ in $\mathbb{Z}$. If $a=\square$, then $\exists d \in \mathbb{Z}$ ...
4
votes
2answers
46 views

$(n+1)^{\textrm{st}}$ prime less than $2^{2^n}$

Using elementary means, show that the $(n+1)^{\textrm{st}}$ prime is less than $2^{2^n}$ please do not use fancier stuff like the prime number theorem or beyond. using this how can you show that ...
0
votes
0answers
26 views

The sets are equal

I want to show that $Z_p^*= \{ x \in Z_p | |x|_p=1 \}$. I have tried the following: Let $x \in Z_p^*$, then $x=a_0+a_1p+a_2p^2+ \dots \ \ \ \ \ \ \ \ 1 \leq a_i \leq p-1$. So, $x \in Z_p $. When ...
0
votes
0answers
28 views

Identity of the $p-$norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
0
votes
0answers
24 views

The representations of numbers by decimals

I'm looking for books that talk about the representation of the integers by decimals, more specifically for prime numbers. I can't found anything yet, I read something in "AN INTRODUCTION TO THE ...
1
vote
0answers
15 views

Prove identities-p norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
1
vote
4answers
103 views

Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?
0
votes
2answers
59 views

A curious elementary number theory problem

Find all $n$ satisfies $\forall k ((k,n)=1$ $\Rightarrow k^2 \equiv 1 (\mathrm{mod} n))$. For example when n=8, k=1,3,5,7 satisfies the condition. When n=24,k= 1,5,7,11,13,17,19 satisfies the ...
1
vote
1answer
35 views

to count the intervals

A finite set of two or more consecutive natural numbers is called a "co-prime interval" if there is no number in it that is co-prime to all other numbers in the set. Given a range [A, B], I would ...
0
votes
1answer
55 views

How can we show that it is an integer 5-adic number?

Show that the number $\frac{3}{8}$ is an integer $5$-adic and calculate the first five positions of its power series in $\mathbb{Q}_5$. Could you explain me how we can conclude that $\frac{3}{8}$ is ...
1
vote
1answer
22 views

If $n = m^2 + 1$ and $x$ is a square modulo $n$, then how to show that $n - x$ is also a square modulo $n$?

I see that if $x \equiv y^2 (\text{mod } n)$, then $n - x \equiv m^2 - y^2 + 1 \equiv (m+y)(m-y) + 1 (\text{mod } n)$. However, I'm not sure how to proceed from there. I'm a complete beginner at ...
4
votes
2answers
80 views

Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?

I believe I see that $a_n = 2^n(a_0+1) - 1$ but am somewhat uncertain where to proceed afterwards. I am a complete beginner at number theory and would appreciate it if someone could point me in the ...
0
votes
0answers
17 views

Conductor of Dirichlet character divides every quasiperiod

Let $\chi$ be a Dirichlet character which has quasiperiods $d_1, d_2$. I.e., if $(n(n + kd_i), q) = 1$ then $\chi(n + d_i) = \chi(n)$ for any $k \in \mathbb{Z}$. Supposedly we can then show that ...
1
vote
1answer
11 views

How to show the congruence involving the order

If $\alpha$ is the lower positive integer such as $x^{\alpha}\equiv 1\mod m$ and the order of $x$ in modulus $m$ is define by $\alpha = \textrm{ord}_{m} x$ Prove that : $$a \cdot b \equiv 1\mod m ...
1
vote
1answer
50 views

Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations, $x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?
-1
votes
1answer
74 views

Check if $N$ is of form $6A + 8B$

Given a number $N$ we need to check if its of form $6A + 8B$ .If its of this form then we need to check if $B$ can be greater than equal to $1$ or not. Like $24$ is of form $6A + 8B$. Also $B$ can ...
0
votes
2answers
51 views

Which of the following is true?

Let $\hspace{0.2cm}$$p,q,r$$\hspace{0.2cm}$ be prime numbers greater than 100,then which of the following is true? $3|p^2+q^2+r^2$ $q|p^5$ There exists integers $x,y$ such that ...
3
votes
1answer
29 views

Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$

Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$. Can we have $$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$ for some constant ...
0
votes
0answers
23 views

Euler's Theorem when $m$ is square-free

Suppose that $m$ is square-free, and that $k$ and $\bar{k}$ are positive integers such that $k\bar{k} \equiv 1\pmod{\phi(m)}$. Show that $a^{k\bar{k}} \equiv a \pmod m$ for all integers $a$. In the ...
0
votes
3answers
32 views

Canonical decompositions and product of primes

Let $S$ be the set of natural numbers $n$ that have exactly $9$ positive divisors. Describe all possible canonical decompositions (as products of primes) of elements of $S$.
2
votes
1answer
66 views

Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is integer

As stated in title, I would like to find solution to this problem: Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is also integer. I need idea how to solve ...
1
vote
2answers
35 views

Homework gcd. Show that $gcd(a_1,a_2,a_3,…a_k) = gcd(gcd(a_1,a_2),a_3,…a_k)$

Help me with this please show that $gcd(a_1,a_2,a_3,...a_k) = gcd(gcd(a_1,a_2),a_3,...a_k)$ How can I start?
2
votes
1answer
52 views

What is the maximum difference between two successive real numbers in the given floating point representation?

The following is a scheme for floating point number representation using 16 bits. Sign :- Bit 15 Exponent:-Bit 14-9 Mantissa :- Bit 8-0 Let $s, e,$ and $m$ be the numbers represented in binary in ...
0
votes
1answer
23 views

Combinations of sets raised to the power of a prime modulus

This is a problem out of the text Introduction to the Theory of Numbers by Niven, Zuckerman, and Montogmery and I am having quite a bit of trouble with it. I tried to prove it directly, but that ...
3
votes
0answers
49 views

Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?

Pythagorean triple is a triple of integers $(a, b, c)$ such that $a^2+b^2=c^2$. Is there any Pythagorean triple such that, not only $a^2+b^2$, but also $b^2+c^2$ is a square number? If not, how to ...
1
vote
4answers
57 views

$z^2=x^2+y^2$ Prove that $4\mid xyz$ ($xyz$ is divided by $4$)

$z^2=x^2+y^2$ where $x,\ y,\ z$ - integers Prove that $4\mid xyz$ ($xyz$ is divided by $4$) All possible rest in divided by $4$ in this case is $1$. That's all I noticed.
1
vote
1answer
54 views

Find all positive integer that $2^{2^n}+5 $ is a prime number. [duplicate]

Find all nonnegative integer that $2^{2^n}+5 $ is a prime number. For $n=0$ we have $7$ - correct For $n=1$ we have 9 - false For $n=2$ we have 21 - false For $n=3$ we have 259 ... Maybe any ideas ...
-2
votes
1answer
50 views

Find minimum possible area of brush

A rectangular brush has been moved right and down on the painting. Consider the painting as a $n × m$ rectangular grid. At the beginning an $x × y$ rectangular brush is placed somewhere in the frame, ...
0
votes
1answer
51 views

Can we not apply the Hensel Lifting Lemma in this case?

Check if the equation $x^2=-1 \text{ in } \mathbb{Z}_2$ has a solution, and if it has, calculate the three first positions of the solution. So, we are looking for a solution $\pmod 2$, one solution ...
0
votes
1answer
80 views

Distance between powers of 2 and 3

As we know $3^1-2^1 = 1$ and of course $3^2-2^3 = 1$. The question is that whether set $$ \{\ (m,n)\in \mathbb{N}\quad |\quad |3^m-2^n| = 1 \} $$ is finite or infinite.
1
vote
3answers
55 views

Finding if a number is prime by looking at the sum of their digits

Take a number $N = \overline{abcdef...}$ where $a, b, c, d,e,\dots$ are the digits of $N$. Let $k$ be the sum of those digits : $a+b+c+d+e+... = k$ If $k$ is any of ${1, 2, 4, 5, 7, 8 }$ then $N$ ...
1
vote
0answers
62 views

IMO 1983 Solution - Day 1 Problem 3

The questions goes as follows: Let $a$ , $b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc - ab - bc - ca$ is the largest integer which cannot ...
0
votes
1answer
59 views

Prove all multiples of $U$ contain all the digits $0$ to $9$

I have to prove that the number $U = 5263157894736842101$ is a "constant number" (that is, every positive multiple of this number contains all the digits from $0$ to $9$ at least one time). In ...
1
vote
1answer
34 views

Number of different vectors.

Let's say that I have a vector with 6 elements. I put two wedges in the vector, i.e., at position 2 and position 6, for instance. And when I say put a wedge, it means... for every time you traverse ...
4
votes
1answer
60 views

A number theory puzzle

I recently came across the following number theoretic puzzle. Suppose I've infinitely many cards, each with a natural number written on it. Given any $n\in \mathbb N$, the number of cards which have a ...
1
vote
0answers
81 views

Why is $2^{16} = 65536$ the only power of $2$ less than $2^{31000}$ that doesn't contain the digits $1$, $2$, $4$ or $8$ in its decimal representation

$65536$ is the only power of $2$ less than $2^{31000}$ that does not contain the digits $1$, $2$, $4$ or $8$ in its decimal representation. http://en.wikipedia.org/wiki/65536_%28number%29
2
votes
1answer
61 views

Closed form for $\sum_1^\infty 1/p^n$

I was wondering if there are some studies on closed forms for the sum $$\sum_{p \in \mathbb{P}}^\infty \frac{1}{p^n},$$ where $\mathbb{P}$ denotes the set of prime numbers. Obviously I know that ...