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

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Moebius Identity

Is there alternative proof of Moebius identity i.e. sum of moebius function over divisor of n is zero than as suggested n page: ...
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0answers
11 views

Dirichlet product is associative

Is there alternative proof of fact: Dirichlet product on arithmetic function is associative than given in Dirichlet's product with number theoretic functions
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1answer
15 views

Every collection of periodic sets $A_n \subset \Bbb{N}$ (minus a common point), that avoids…

Let $\{A_n\}$ be a set of subsets of $\Bbb{N}$ each of which are periodic except for a common point. That is to say, there exists one and only one $x_0$, such that for each $n$, if $x \in A_n, x \neq ...
6
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3answers
57 views

Diophantine equation $(x+y)(x+y+1) - kxy = 0$

The following came up in my solution to this question, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation $$ (x+y)(x+y+1) - kxy = 0$$ For $k=5$ ...
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1answer
32 views

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime.

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime. Though I didn't check the statement with so many integers, but it looks like the equation never ...
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1answer
35 views

How to prove the sum of squares larger than 1/n without induction? [duplicate]

known that: $1\geq R_1 \geq R_2 \geq \dots \geq R_n \geq 0$ and $\sum_{i=1}^n R_i=1$ To prove: $\sum_{i=1}^n R_i^2 \geq \frac{1}{n}$ Using induction, the problem can be easily proved. I'd like to ...
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2answers
26 views

Number 9 and age of mother when child is born.

If a mother's age is divisible by 9 when a child is born then once you go to the next decade,n every 11 years the child's age and mother's age are always the same two numbers in reverse order. For ...
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22 views

Prove that for every $k$ there exist fibonnaci number that ends with $k$ zeros.

Let $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Prove that for every $k$ there exist $F_m$ that ends with $k$ zeros. I tried using pigeonhole principle, but with no effect.
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2answers
52 views

“Proof” that $(2n)!$ is divisible by $2^n 5^{n-3}$ for $n\ge3$

Please explain, as clearly as possible, what is wrong with the following "proof" by induction that $\hspace{1.4 in}$$(2n)!$ is divisible by $2^n 5^{n-3}$ for $n\ge3$. (There clearly must be an ...
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2answers
39 views

4th Grade Common Core question [on hold]

In the number $2,119$ if you move from the $1$ in the hundreds place to the $1$ in the tens place, what happens to the value of the $1$?
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14 views

Lehmer's totient problem generalization (adding a constant )

Lehmer's conjecture is an well-known open problem which states that the divisibility : $$\phi(n) \mid n-1$$ holds only for primes . This motivated me to ask the following : For which ...
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1answer
37 views

Solving problem of abstract algebra [on hold]

The question is that if $n$ is not a multiple of 23 then the remainder when $n^{11}$ is divided by 23 is 1 or -1(mod 23). Is it true or false? Please answer me.
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0answers
63 views

Solving an equation $x^{22}\equiv2 \bmod 23$ [on hold]

I have an abstract algebra problem which I am unable to solve. The problem is, if $x^{22}\equiv2 \bmod 23$, then $x$ has how many solutions? Please explain me.
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1answer
43 views

Maximum of $xy^3z^7$ in the plane $x+y+z=1$

A friend gave to me this problem and on having seen that I could not solve it in the first instance helped me with the hint of using the AM-GM inequality. PROBLEM.- To maximize the product $xy^3z^7$ ...
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0answers
35 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
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1answer
37 views

Is this a good generating function for Sum-of-divisors function?

I have an expression for the sum-of-divisors function defined as $$\sigma(n)=\sum_{d\mid n}d.$$ However I do not know how nontrivial or practical it actually is. Let us define ...
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1answer
21 views

Randomly picking 2 integers to compute a third one with equiprobability

I have a problem, that might be simple but I just don't see it for the moment. Supposing you have a finite set of integers $S_1$, I am looking for a simple function that when randomly picking two ...
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1answer
18 views

If $a, b \mid c \text { and } \gcd(a, b) = d, \text { then } ab \mid cd $

$a \mid c \to c = ak \text { and } b \mid c \to c = bj.$ $ak + bj = 2c = d \to c \mid d.$ $d \mid a \to a = dj.$ $c = ak = d(jk) \to d \mid c.$ So, $c = d.$ $a \mid c \text { and } b \mid c ...
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0answers
42 views

Is it true that this number is not prime?

Let $p=2^{4n}+1$. It follows that $p\equiv2,3,5 \mod{7}$. If $p$ is prime can we do better? I mean is it true that if $p\equiv 2 \mod{7}$ then $p$ is not prime? This is equivalent to prove that ...
4
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1answer
21 views

Basic question on equivalence relations.

Show that the following relation is an equivalence relation on the given set. $m \sim n$ in $\mathbb{Z}$ if $m \equiv n\,(\text{mod}\,6)$.
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3answers
50 views

Prove or disprove divisibility claims? [on hold]

a) If $x^2$ is a multiple of $4$, then $x$ is a multiple of $4$ b) If $x^3$ is a multiple of $2$, then $x$ is a multiple of $2$
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3answers
919 views

Intermediate digits of 34!

Problem: Given that $34!=295232799cd96041408476186096435ab000000$. Find $a, b, c, d$. $a, b, c, d$ are single digits. I am able to find $a$ and $b$ but cant find $c, d$. I did the prime factorisation ...
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4answers
89 views

Proving $n! = n \Rightarrow (n = 1 \quad or \quad n = 2)$

I want to know whether my proof is correct. Any elegant proofs are welcome. $n\in\mathbb{N}.\quad$Prove $ (n!=n) \Rightarrow (n=1\quad or\quad n=2)$ $ (n!=n) \Rightarrow (n=1\quad or\quad n=2) ...
5
votes
2answers
48 views

Suppose $p, p+2, p+4$ are prime numbers. Prove that $p = 3$ not using division algorithm.

Suppose $p, p+2, p+4$ are prime numbers. Prove that $p = 3$ not using division algorithm. Hint: why can't $p = 5$ or 7? So I have done the two hints and in both cases I get a 9 in my set of numbers, ...
6
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2answers
66 views

Does every prime of the form $4k+1$ divide a number of the form $4^n+1$?

While playing around with Fermat's little theorem I was asking myself the question in the title and I can't answer it...
2
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0answers
18 views

How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
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3answers
111 views

For each positive integer $a$, does there exist a positive integer $b$ such that $2b^2 + b \gt ab^2$?

The problem is this: Prove or give a counterexample to the following statement. For each positive integer $a$, there exists a positive integer $b$ such that $2b^2 + b \gt ab^2$. I've tried a couple ...
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4answers
65 views

Show that there is a number on the form $11 \dots 000 \dots 0$ divisible by 2014

Show that there is a number on the form $11 \dots 000 \dots 0$ (some number of $1$s followed by $0$s) divisible by $2014$. I'm helping someone practise for the math olympiad, and this question has me ...
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1answer
38 views

Find the natural number $n>2$ such that $\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$ [on hold]

I'm unsure how I'm supposed to solve the equation: $$\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!} $$ given that $n>2.$
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3answers
82 views

How do you simplify $n!-(n-1)!$ [on hold]

I'm unsure how to simplify the expression $n!-(n-1)!$. Working as well as the final answer would be preferable.
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3answers
1k views

Can 720! be written as the difference of two positive integer powers of 3?

Does the equation: $$3^x-3^y=720!$$ have any positive integer solution?
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1answer
71 views

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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1answer
21 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
45 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
39 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
23 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
68 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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1answer
31 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
33 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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0answers
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
28 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 ...
2
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0answers
31 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 ...
3
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2answers
63 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 ...
3
votes
2answers
114 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 ...
2
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1answer
102 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
24 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, ...
0
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1answer
30 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 ...
5
votes
1answer
55 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 $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$