Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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1answer
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Proving congruence class

Let $a$ and $m$ be integers such that $m ≥ 1$. Consider the congruence class of $a$, $[a]$ modulo $m$. It follows that $∀ x ∈ [a], \gcd(x, m) = \gcd(a, m)$. I have my algebra midterm in two ...
2
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0answers
33 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
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1answer
26 views

The primes such that removing digits from the right end leaves another prime

The number 73,939,133 is prime. Keep removing a digit from the right end. Each of the remaining numbers is prime. How to find other numbers with this property?
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0answers
9 views

Does there exist a sequence $(S_i)_{i=1}^{\infty},\ S_i=\pm1$ such that $\forall i(2+S_1g_1+S_2g_2+\cdots+S_ig_i\in\Bbb P)\wedge\exists i:S_i=-1$?

Consider a sequence $(S_i)_{i=1}^{\infty},\ S_i=\pm1$ other than $\{1,1,\ldots\}$. Let $g_i=p_{i+1}-p_i$, where $p_i$ is the $i$th prime. Is it possible that for all $k\in\Bbb Z^+,\ ...
2
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1answer
19 views

Coprime numbers and equations

Suppose $~m~$ and $~n~$ are coprime and both of them are greater than one. Is it right that equation $~mx + ny = (m-1)(n-1)~$ has solutions over non-negative integers? For example $~ (x,y) = (6,0) ...
6
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0answers
39 views

Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
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2answers
38 views

For every prime of the form 6x-1 are there comparable number of primes of the form 6x+1

All primes except $2$ and $3$ are of the form $6x-1$ and $6x+1$. For every prime of the form $6x-1$ are there comparable number of primes of the form $6x+1$ in the first $10000$ primes or is there an ...
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0answers
22 views

Getting a decomposition of an integer

Firstly, I need to admit that my English is quite poor. Ok. I've got a problem- how can I get any possible decompositions of a given integer? Sample decomposition: $$\begin{align} ...
1
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1answer
24 views

Let $\{A_n\}_{n=1}^\infty$ be a sequence of nonempty subsets of $\mathbb{Z}$, which of the following is uncountably infinite?

Let $\{A_n\}_{n=1}^\infty$ be a sequence of nonempty subsets of $\mathbb{Z}$. Which of the following is uncountably infinite? A) $A_1$ B) $\bigcap_{n=1}^\infty A_n$ C) $\bigcup_{n=1}^\infty A_n$ ...
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2answers
31 views

Compute the remainder of a division [on hold]

Could you help me about this easy problem. I have to compute the remainder of the division of $9^{123456789} $ by 17 Thanks!
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0answers
37 views

The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function? 2 - How to generalize the following? Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...
1
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1answer
26 views

Discrete Math Proof Method

Give a direct proof of the fact that $a^2-5a+6$ is even for any integer $a$. Suppose $a$ and $b$ are integers and $a^2-5b$ is even. Prove that $b^2-5a$ is even.
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3answers
854 views

Largest prime number with all digits different

What is the largest prime with distinct digits? (It is certainly less than ten digits long.Can you explain it why?
3
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0answers
15 views

Prerequisites: Dirichlet Lectures Number Theory

I am interested in getting Dirichlet's Lectures in Number Theory but I'm afraid I don't know that much advanced math. Do I need to know things like Determinants for this book? Any list of ...
1
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1answer
64 views

Number of $0$ in the end of $11^n-1$

$n$ is integer, calculate number of $0$ in the end of $11^n-1$(i.e. largest integer $m$ such that $10^m|11^n-1$). The original question was $n=100$ and I could only choose $m$ from 1 to 5. I ...
2
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0answers
43 views

Prove that exist set $B$: $|B|\ge 2014$ .

For $A$ is a set has $2014$ natural numbers. Prove that exist a set $B\subset \mathbb{N}$ such that $A\subset B$ and sum square of all elements of $B$ equal area of all elements of $B$. I think we ...
2
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1answer
35 views

An equation which has solution modulo every integer

In the book Abstract Algebra by Dummit and Foote he remarks that there is an equation which has solutions modulo every integer but has no integer solutions. The equation he gives is ...
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2answers
42 views

if n is composite n divides (n-1)!

We need to prove that if n is a composite number >4, then n|(n-1)!. I wanted to ask if my observation is correct or not..what i think is that the statement can be reduced to n|(n-2)! Because n and n-1 ...
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2answers
52 views

Find the number of natural numbers less than 2014 which are neither squares nor cubes [on hold]

Find the number of positive integers less than 2014 that are neither squares nor cubes.
3
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1answer
28 views

A finding (?) about some primitive Pythagorean triples

I have just stumbled on the fact that the sum of the three absolute differences between each pair of a primitive Pythagorean triple [absolute values of (a-b), (b-c) and (c-a), where a,b,c constitute ...
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1answer
28 views

Proving property of congruence - help needed

Let $c,d,m,k ∈ \mathbb{Z}$ such that $m ≥ 2$ and $k$ is not zero. Let $f = \gcd(k,m)$. If $c \equiv d \pmod m $ and $k$ divides both $c$ and $d$, then $$ \frac{c}{k} \equiv \frac{d}{k} ...
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2answers
27 views

The sum of n numbers that cube to one is congruent modulus three.

Assume $a_1,\dots,a_n\in\mathbb C$ cube to give one. Assume $\sum a_i=\sum a_i^2$. How can we see that $\sum a_i\equiv n(mod3)$? May the sum be different than $n$?
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1answer
21 views

Using Hensel's lemma to solve congruence?

I'm trying to use Hensel's lemma to solve the congruence $$x^3 + x^2 - 5 \equiv 0 \pmod{7^3}$$ I begin by solving $$x^3 + x^2 - 5 \equiv 0 \pmod{7}$$ and observe that $x \equiv 2 \pmod{7}$ is the ...
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3answers
20 views

Example involving the Chinese Remainder Theorem

I am working on a Number Theory book and I have come across the following problem: (Underwood Dudley 2nd Edition Section 5 Problem 3): Solve the system: x $\equiv 3(mod 5)$ x $\equiv 5(mod7)$ x ...
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2answers
39 views

$27 | (2x+1)^2 \implies 2x$ is a multiple of 9?

Found this simple fact in a proof that I was looking up, and am confused as to why it is true: Why does $27 | (2x+1)^2 \implies 2x + 1$ is a multiple of 9?
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3answers
65 views

The sum of all the odd numbers to infinity [duplicate]

We have this sequence: S1: 1+2+3+4+5+6.. (to infinity) It has been demonstrated, that S1 = -1/12. Now, what happens if i multiply by a factor of 2? S2: 2+4+6+8+10+12.... (to infinity). I have ...
1
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1answer
22 views

Show that $p^{q^3+q} = p^2$ (mod $q$)

For two distinct primes $p,q$, show that $p^{q^3+q} = p^2$ (mod $q$). Since $gcd(p,q)=1$, it suffices to show that $pq|p^{q^3+q}-p^2$, since $p$ obviously divides that, but I don't know how to ...
4
votes
0answers
30 views

Find $a$ given some additional conditions

The problem is: If $x+y+z=3$ and $xy+xz+yz=a$, where $a$ is a real number, find $a$ if the difference between the maximum and minimum value of $x$ is $8$. So what I did was use Vieta's equations ...
1
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1answer
15 views

Arithmetic Using Different Bases

If $Feed_{base 8}-Feed_{base 5}=Feed_{base 7}$, then what do the digits $F, e$, and $d$ stand for? So far I have that $d = 5$ and $e = 6$. I think those are correct. However, I am getting stuck on ...
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2answers
29 views

Find the number of three digit numbers with a even number of positive divisor

I guess the question is probably just asking for the number of the three digit composite numbers besides the perfect square. So the question critical to solving the problem is really how to find ...
0
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0answers
8 views

one-to-one correspondence with a set of primitive dirichlet charchter

Let $$\operatorname{Prim}_{N}=\{\xi \mid \xi \text{ a primitive Dirichlet charchter mod } F \text{ with } F\mid N\}$$ and $$\operatorname{Char}_{N}=\{\xi" \mid \xi" \text{ Dirichlet charchter mod } N ...
4
votes
2answers
44 views

If for all $n\in\Bbb{N}, a^n-n$ divides $b^n-n$ then $a=b$.

Exercise: Let $a,b\in\Bbb{N}$, show that if for all $n\in\Bbb{N}, \quad a^n-n$ divides $b^n-n$, then $a=b$. I don't have lot of knowledge on this subject, I am aware about some elementary result ...
0
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1answer
39 views

how to predict the sum of digits of the number $A(n)$ for a large natural $n$ without calculation, when $A(n)=a(n^2+n)+b$?

look $A(n)=9n^2+9n-1$ , let $n=15233$ , $A(15233)=2088535697$ the sum of digits of this obtained number is :$53$ and always take this form :$9k+8$ , where $k=5$ and always exist a natural number $k$ ...
0
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1answer
40 views

Clarification of Legendre's theorem re: $ ax^2+by^2=cz^2$

Theorem (Legendre): Let a,b,c coprime positive integers, then $ax^2+by^2=cz^2$ has a nontrivial solution in rationals x,y,z iff $(−bc/a)=(−ac/b)=(ab/c)=1$. I read this somewhere. Is it really the way ...
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0answers
20 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
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0answers
20 views

Which theorem could be used?

I want to write the $p-$adic expansion of $6!$ in $\mathbb{Q}_3$. I have to solve the congruence $x \equiv 6! \pmod {3^n}$, right? I found the following: $$x_0 \equiv 6! \pmod 3 \Rightarrow x_0 ...
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2answers
37 views

How can I prove that the square root of two prime numbers multiplied is non-rational number?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
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1answer
27 views

Prove the divisors pairs

If we arrange the positive distinct divisors of a number A by increasing order, then we get something like: $$1<a_1<a_2<a_3<...<a_{n-2}<a_{n-1}<a_n<A$$How can we prove that ...
0
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5answers
88 views

Showing $p^2 + q^2\ne r^2$ for primes $p, q, r$. [on hold]

Let $p$, $q$, $r$ be prime numbers. Show that $p^2 + q^2\ne r^2$.
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5answers
68 views

Do there exist integers s and t such that 11s + 9t = 1?

Do there exist integers s and t such that 11s + 9t = 1? We just started learning discrete mathematics and I am absolutely stuck with proof questions. Does this question belongs to number theory ...
0
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1answer
57 views

Find $n$ such that $n/2$ is a square, $n/3$ is a cube, and $n/5$ a fifth power

Consider the set of positive integers $n \in \mathbb {Z}>0$ such that $\dfrac{n}{2}$ is a perfect square, $\dfrac{n}{3}$ is a perfect cube, and $\dfrac{n}{5}$ is a perfect fifth power; that is to ...
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1answer
43 views

Proof of a sum of positive divisors

Let $n$ be an integer greater than zero. Prove $$(\sum_{d|n}v(d)){}^{2}=\sum_{d|n}(v(d))^{3}$$ where $v(d)$ is the number of positive divisors of $n$. I'll outline what my problem is. I write $n= ...
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2answers
91 views

Two math professors problem

My friend asks me a question from internet. The question is as follows Two math professors, professor Uno and professor Dos, play chess at the park while reminiscing about their past. Prof. ...
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2answers
17 views

Finding the closed form of a multiplicative function

Let $n$ be an integer and $n>0$. Define function $g$ by $g(1)=1$ and $g(n) = 2^{m}$, where $m$ is the number of distinct prime numbers in the prime factorization of $n$. I've already proven that ...
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6answers
57 views

How to know a number is divisible by a given number without using a calculator?

My question is simple and comes from my curiousity during studying math. How to know a number is divisible by $7$ or $13$ without using a calculator? For example, how do we decide intuitively that ...
2
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0answers
46 views

integral solutions of $ ax^2+by^2=c$ [on hold]

Let $a,b,c,x,y$ be all non-zero positive integers, $\gcd(a,b,c)=1$, find the integral solutions of:$$ ax^2+by^2=c$$ Any hint?
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2answers
26 views

Show that in the sequence b,2b,3b…,mb there are exactly gcd(b,m) numbers divisible by m.

"Where b and m are integers and m is bigger than one, show that in the sequence b,2b,3b...,mb there are exactly gcd(b,m) numbers divisible by m. I'm having a real hard time proving that... can anyone ...
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0answers
27 views

Fundamental Theorem of Arithmetic (Canonical) missing crucial step

I've worked long on the proof of the fundamental theorem of Arithmetic and there is only one tiny piece left I can't wrap my head around. Suppose that $$\prod_{i=1}^r p_i^{m_i} = \prod_{j=1}^s ...
2
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0answers
19 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n ...
4
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0answers
56 views

Floor function inequality $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$

Let $a,b$ be positive integers. Show that $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$. [Source: Russian competition problem]