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

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Prove that $s(n-1)s(n)s(n+1)$ is always an even number

Let $n$ be a natural number, and let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n>1$ the product $s(n-1)s(n)s(n+1)$ is always an even number. I calculated the sum of ...
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16 views

Divisors and non divisors

How do i prove: The sum over the numbers that don't share a divisor with the divisors of n, including n it self, will be n-1. Example $n=12$: The numbers that share a divior are $2;3;4;6;12$. ...
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1answer
15 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
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2answers
43 views

Weird question about natural numbers. Obvious or not?

Given any subset $A,C \subset \Bbb{N}$, there exists a maximal subset $B \subset \Bbb{N}$ such that for all $b \in B, a \in A, \ |b - a| \in C$. For instance $A = \{3,5\}$, $C = \{2,4\}$, then ...
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1answer
35 views

If $p$ is prime and $p\equiv 3 \pmod 5$, show that for every $a$, $x^5\equiv a \pmod p$ is solvable.

I tried all sort of things. I know it is supposed to be easy but I can't seem to be thinking anymore. I could really use even the most basic lead here. I tried working with primitive roots and ...
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1answer
22 views

Does there exist a finite set of homogeneous polynomials (+ property) whose unique solution is equivalent to a finite sequence of naturals?

Consider the set $\{2,3,5\}$ of natural numbers. Letting $p = 2, q = 3, r = 5$ we have: the polynomial equations: $$p + q = r, \\ p^2 + r = q^2 \\ q^3 - p = r^2 $$ Each is a homogeneous ...
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2answers
53 views

In the $x + 1$ problem, does every positive integer $x$ eventually reach $1$?

I know that the more famous $3x + 1$ problem is still unresolved. But it seems to me like the similar $x + 1$ problem, with the function $$f(x) = \begin{cases} x/2 & \text{if } x \equiv 0 ...
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0answers
25 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
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1answer
30 views

Find all primes $p$ for which $x^2+2x+4\equiv 0 \pmod p$ is solvable. Am I correct?

Getting ready for an exam, I would like to focus on the correctness of my solution, final results and assumptions, and would appreciate any comment regarding it or even additional ...
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18 views

The distribution of prime and semi-prime.

Let $\alpha$ be an integer and $\rho_1,\rho_2$ some prime such that $\alpha=\rho_1\cdot\rho_2+1$, and $\beta$ the number of all semi-prime less than or equal to $\alpha$. Prove ...
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2answers
28 views

Proving that $a^{b}$ is rational (Elementary number theorey) [duplicate]

Prove that there exist irrational numbers $a$ and $b$ such that $a^{b}$ is rational. What i tried Prove by contradiction I assume the statement For all rational numbers $a$ and $b$ such that ...
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0answers
25 views

Showing $pk+1|p^p-1$ implies that $k$ is even

Suppose $p$ is an odd prime such that $pk+1$ divides $p^p-1$. Prove that it is not possible for $k$ to be odd. Here's my solution: Assume to the contrary that $pk+1$ does divide $p^p-1$ We can ...
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0answers
28 views

$a$ and $n$ are relatively prime then there exists an integer $k$ such that $ak \equiv 1(\mod n) $.

Let $n \in Z$, $n > 1$ and let $a \in Z$ with $1 \leq a \leq n$. Prove that if $a$ and $n$ are relatively prime then there exists an integer $k$ such that $ak \equiv 1(\mod n) $. Proof: Suppose ...
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1answer
25 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
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2answers
60 views

Prove that for every natural number $n > 2$ there is a prime number between $n$ and $n!$

So I have already read this page with the solution: For all $n>2$ there exists a prime number between $n$ and $ n!$ Now I was able to reason that $p < n!$ Because I was given the hint that ...
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2answers
112 views

Perfect numbers

Define a Perfect (capital-P) number as a natural number that is equal to the sum of its Divisors excluding 1 and the number itself. (So the Divisors of 28 are 2, 4, 7, 14, summing to 27.) Is there any ...
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1answer
62 views

Difficult sets of Equations, counting

Let $ m$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2000$. Find the ...
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0answers
21 views

Is this an Alternate proof of the Uniqueness of Remainder Th.?

$\mathbf{Theorem}: \forall$ pairs of integers $(a,b)$, $\exists$ unique integers $q, r$ such that $a=bq+r$ and $0\le r \lt |b|$ Assuming that the existence of the integers $q,r$ is already proven, ...
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1answer
29 views

If $n=x^2+3y^2$ then any prime in $n$'s factorization is of an even power.

If $n=x^2+3y^2$ then any prime $p$ such that $p\equiv 2 \pmod 3$ in $n$'s factorization is of an even power. I have been spending hours trying to solve this because of some some issues withholding any ...
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1answer
53 views

Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$

Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$
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Smallest number of workers in factory, Diophantine approximation

Q. In a factory, the percentage of male workers was $53.7802\%$ (rounding to nearest fourth decimal place) last year. What is the smallest number of female workers working there? Hint: Diophantine ...
2
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1answer
31 views

$(a^{2^n}+1,a^{2^m}+1)=1 or 2$ [duplicate]

Prove that if $m\not =n,a$ are positive integers then $(a^{2^n}+1,a^{2^m}+1)$ is $1$ if $a$ is even and $2$ if $a$ is odd. I solve the problem in the following way: I assume that $m>n$ then ...
4
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1answer
54 views

Is every sufficiently large even integer the sum of distinct primes?

Is every sufficiently large even integer the sum of (any number of) distinct primes? No doubt this question has been asked before; does the conjecture/theorem have a name? It is related to ...
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1answer
66 views

Integer solutions to $ab=a+b$

If $ab=a+b$ is there only one possible solution,i.e. $a=b=2$? ($a$ and $b$ not equal to zero and are integers). If not what are the others? I have proved that $a$ always needs to be equal to $b$. My ...
5
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0answers
41 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
3
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2answers
510 views

A number n is not a Prime no and lies between 1 to 301,how many such numbers are there which is not divisible by 2,3,5,7.

A number n is not a prime and lies between $1$ and $301$, how many such numbers are there which are not divisible by $2$, $3$, $5$, $7$? a) $6$ b) $8$ c) $10$ d) $12$ My approach: ...
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2answers
39 views

If $\gcd(ab,c)=d$ and $c|ab$ then $c=d$

For all positive integers $a$, $b$, $c$ and $d$, if $\gcd(ab, c) = d$ and $c | ab$, then $c = d$. Need help proving this question, I know that $abx + cy = d$ for integers $x,y$ and that $c|ab$ can be ...
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1answer
50 views

Property for the natural numbers.

This question is inspired by another question I had, where I wanted to prove something about the natural numbers. Often in analysis books I see some proofs, where they use the natural numbers, but it ...
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1answer
16 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
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2answers
76 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...
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1answer
35 views

For every natural integer $N>3$ there are at least two distinct prime numbers $p$ and $q$ such that $\dfrac{p+q}{2}=N$ and $N-p=q-N$, $(p<q)$.

I'm not sure but this problem may be similar or related to Goldbach conjecture? Any proof/disproof, insight and opinion is appreciated, thanks.
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1answer
35 views

Is it impossible to recover multiplication from the division lattice categorically?

In this question it was asked if the division lattice (i.e., the preorder category $(\Bbb Z_{>0}, \mid)$) contains enough information categorically to recover the relation ...
5
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0answers
45 views

Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which ...
3
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2answers
81 views

New deterministic primality test for numbers of the form $p\cdot 2^n + 1$

Edit: Sorry, there was an error. Old Claim (not true because there is a counter-example): Let $p$ be prime. Let $n \in \left\{1, 2, 3, ...\right\}$. Then $N = p\cdot 2^n+1$ is prime, if and only ...
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1answer
60 views

Each number in $S\subseteq \{1,\ldots,2n\}$ does not divide another one, with $|S|= n$. In how many ways?

Let $f(n)$ be the number of subsets $S\subseteq \{1,2,\ldots,2n\}$ such that $|S|=n$ and $a$ does not divide $b$ whenever $a,b \in S$ are distinct. Can we evaluate $f(n)$, at least asimptotically? ...
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2answers
70 views

Question about irrationality proof of $\sqrt{n}$

I'm talking specifically about a proof that I've found. I don't seem to get some parts of it. It states that if you take: $$\sqrt{n}=\frac{p}{q} \:\: \;\;p,q \in \mathbb{Z} $$ where $p$ and $q$ share ...
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0answers
17 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
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1answer
20 views

What am I doing wrong here? Showing $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$.

Show $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$ where $a$ is invertible. What I did is: If $\text{Ord}_{N}(a)|k$ it is obvious. Suppose $a^k\equiv 1 \pmod N$. Not let us assume by contradiction ...
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1answer
12 views

What place value rounded would be the closest answer to the difference between 62960 and 49605 [on hold]

What place value rounded, ten thousands, thousands or hundreds would be the closest answer to the difference between 62960 and 49605
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2answers
25 views

The number of ordered pairs of positive integers $(a,b)$ such that LCM of a and b is $2^{3}5^{7}11^{13}$

I started by taking two numbers such as $2^{2}5^{7}11^{13}$ and $2^{3}5^{7}11^{13}$. The LCM of those two numbers is $2^{3}5^{7}11^{13}$. Similarly, If I take two numbers like ...
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1answer
34 views

exercise Question-29 from contemporary abstract algebra [on hold]

Consider the element A=(1101) in SL(2,R) what is the order of A? If we view A=(1101) as a member of SL (2,Zp), what is the order of A
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1answer
26 views

Continuous functions with domain in the Natural Numbers

Can functions with domain in the Natural Numbers be continuous? In the high school, it is teached an intuitive notion of continuous functions: functions which will always appear as an "unbroken ...
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1answer
66 views

If $a^{n}-1$ is divisible by $n$, then $a+1,a^2+2, \ldots a^n+n$ leave different remainders when divided by $n$

It is given that $a^n-1$ is divisible by $n$ for some natural numbers $a$ and $n$. Prove that the numbers $$a+1,a^2+2, \ldots a^n+n$$ all leave different remainders when divided by $n$. I tried ...
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1answer
15 views

How many ways are there to express a natural as a sum of 3 others—but by induction?

I have figured out an (inductive?) process, but I cannot express it formally: There is always one possibility where $n$ is in the first place of our 3-tuple: $[n~~0~~0]$. Then I can subtract $k~(\leq ...
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1answer
21 views

An equation of three variables has integral solution

The given equation is $63x+70y+15z=2010$ and I have to whether it has any integral solution and if yes what that is . Now this is one equation in $3$ variables . How can ...
3
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1answer
66 views

Each number in a subset $S\subseteq \{1,\ldots,2n\}$ does not divide another one. Then $\max |S|$?

This problem comes from a seemingly innocuous question from a professor during a lesson for a Math Olympiad course. [A part of this question is really a classic of number theory/combinatorics] ...
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4answers
179 views

Find the least positive residue of $10^{515}\pmod 7$. [on hold]

I tried it, but being a big number unable to calculate it.
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3answers
49 views

Solve the congruence $31x\equiv 5 \pmod{23}$

I've used the Euclidean Algorithm to solve congruences of the form $$ax \equiv b \pmod n$$ where $n >a$, for example: $16x \equiv 5 \pmod{29}$. When $n <a$, for example, $$31x \equiv 5 ...
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2answers
32 views

For a natural number b, N(b)= number of natural numbers a such that the equation x2+ax+b=0 has integral roots.

For a natural number $b$, $N(b)= $ number of natural numbers $a$ such that the equation $x^2+ax+b=0$ has integral roots. What is the lowest possible value of $N(6)$?
3
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0answers
59 views

Connections between Fibonacci and natural numbers

Here are some known facts about Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem . For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of the ...