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

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1answer
25 views

Define recursively the sequence of the cubes of natural numbers

Let $n $ be an element of the natural numbers, and let $s(n) $ be the series defined by the squares of the natural numbers, i.e. $s(n) =0, 1, 4, 9, 16, 25, 36,... $ I have worked out the recursive ...
3
votes
1answer
38 views

Frazzle game question

In $7^{th}$ grade, in order to learn divisibility, memory, and focus, my math teacher had my pre-algebra class play a game called Frazzle. To play the game Frazzle, each person went around the room ...
-2
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1answer
69 views

Can anyone Find the error below [duplicate]

If: $$S=1+2+4+8+....+2^n +...$$ So we get $$2S=2+4+8+...$$ $$2S+1=1+2+4+8+16...$$ $$2S+1=S$$ $$2S-S=-1$$ $$S=-1$$ Is there error, and if there's, why? I want athletic explanation.
0
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1answer
41 views

Show that a prime p divides $a^p - a$. [duplicate]

Let $p$ be a prime. I want to show that $p | a^p - a$. I want to use induction. I showed that this is true for the case $a = 1$. I'm having trouble with the bridge.
0
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2answers
47 views

Find integer solution of sysem of quadratic equations [on hold]

If: $a,b,c$ positive integers, where $a\geq b\geq c$. such that: $$a^2 - b^2 - c^2 +ab=2011$$ $$a^2 +3b^2 +3c^2 -3ab-2ac-2bc=-1997.$$ Find the value of $a$ I tried, but I got nothing. Source: 2012 ...
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2answers
58 views

Highest power of a prime in the product of consecutive factorials [on hold]

$y$ and $n$ are positive integers. $1!\times2!\times3!\times...\times26! = y\times13^n$ $n$ is equals ? ($n$ is above)
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2answers
36 views

Comparison of two sets of 4-tuples using combinatorics

My problem is to show that $\mathbf{A} = \mathbf{B}$. Specifically that $\forall a \in \mathbf{A} \implies a \in \mathbf{B}$ and $\forall b \in \mathbf{B} \implies b \in \mathbf{A}$, to be precise. ...
2
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2answers
61 views

Prove that $2^{9693}-1$ divisible by $7$

Prove that $2^{9693}-1$ divisible by $7$, by more than one way. my try... that, the power divisible by $3$ so it's divisible by $7$ like $2^3,2^6,2^{12}$ and I think it's wrong.
0
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1answer
47 views

Please help me understand this modular arithmetic.

We have this question on a review sheet in my discrete math class and I have also provided the given answer. I am totally lost when it comes to this stuff, and I would just like an explanation of why ...
3
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2answers
46 views

Three variable, second degree diophantine equation

I am trying to solve this diophantine equation: $x^2 + yx + y^2 = z^2$ In other words, I am trying to find integers $x$ and $y$ such that $x^2 + yx + y^2$ is a perfect square. So far, the only ...
6
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1answer
75 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
0
votes
2answers
33 views

If $g$ is a primitive root modulo $p$, then $(p-1)! \equiv g^{p(p-1)/2} \pmod{p}$

Does anyone know how to prove the following theorem: If $g$ is a primitive root modulo $p$ (and $p$ is a prime number), then $$(p-1)! \equiv g^{p(p-1)/2} \pmod{p}.$$
2
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0answers
38 views

Calculation of product of all coprimes of number less than itself

Is there any fast way or formula to calculate product of all coprimes of a number less than itself? How can we do it without finding all coprimes manually?
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0answers
51 views

prime number with property same as 653 [on hold]

The prime number $653$ is a prime number where $5$ is neither a primitive root nor quadratic residue of $4N^2+1$. And I checked primes up to $1000000$ but I couldn't find a single prime number with ...
2
votes
1answer
21 views

Prove zeta-esque relationship with floor function

I'm looking to show that: $$ \frac1{1-2^{-s}} \frac1{1-4^{-s}} = \sum_{k=0}^\infty \frac{\left\lfloor1+\frac{k}{2}\right\rfloor}{2^{ks}}$$ I've noticed $(1-4^{-s})^{-1} = (1-2^{-2s})^{-1}$ and I'm ...
7
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2answers
84 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
0
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0answers
9 views

Adam isomorphism of circulant graphs

Let $C(n; S)$ denote a circulant graph on $n$ vertices (the vertices can be labeled $0,\ldots,n-1$), and connection set $S = \{s_1, \ldots, s_k \}$. Let $1 \leq \mu < n$ be relatively prime to $n$. ...
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0answers
17 views

A question on the representation of all integers in terms of the sum of other interger cubes [duplicate]

The question is from a book used for transition between high school mathematics and university mathematics, which states: Prove the following statement or give a counterexample $\forall n \in ...
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3answers
44 views

Find all numbers that have 30 factors and have 30 as one of their factors.

Find all numbers that have 30 factors and have 30 as one of their factors. Thank you. Note: please show way if possible.
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1answer
27 views

The solutions to $x^m \equiv 1 \bmod p$ will all be solutions to $x^{mn} \equiv 1 \bmod p$ for any $n$.

Can you please show the proof of "The solutions to $x^m \equiv 1 \bmod p$ will all be solutions to $x^{mn} \equiv 1 \bmod p$ for any $n$."
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0answers
29 views

Solve $x^n\equiv1 \pmod p$ for $x$ where $n$ is odd, $p$ prime [duplicate]

The solution to $x^3\equiv1 \pmod p$ has been discussed in Solve $x^3 \equiv 1 \pmod p$ for $x$ and explained elegantly by Arturo Magidin. The discussion established the form of $p$ and $x$. What ...
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2answers
67 views

Find all integer numbers $n$ such that $\frac{11n-5}{n+4}$ is a perfect square.

Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$ I really tried but I couldn't guys, help please.
2
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1answer
42 views

Quantity of elements of order $d$ in $Z_n$, with $d \mid n $

I'm studying for an exam and I can't answer this problem. I'd appreciate a hint. What I've got so far: Let $x$ be an element or order $d$. Then, $x\cdot d \equiv 0 \pmod n \Rightarrow n \mid x\cdot ...
1
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1answer
29 views

Comparing two statements of Chinese Remainder Theorem (Sun-Ze Theorem)

Wikipedia states the Chinese Remainder Theorem as follows: Suppose $n_1, \dots, n_k$ are positive integers which are pairwise coprime. Then for any given sequence of integers $a_1, \dots, a_k$, ...
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4answers
79 views

How do I show that:if$p$ is prime $>5$ then $p^4-20p^2+19$ is always divisible by $180$.?

Is there someone who can show me How do i show that :If $p$ is a prime number greater than $5$ then : $$p^4-20p^2+19$$ is always divisible by $180$. Note : i think should factor $p^4-20p^2+19=$ ...
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1answer
38 views

Maximum number of positive integers $x\neq y$ such that $\frac{xy}{100}\leq|x-y|$

I've been trying to solve the next problem but I have no idea of how to find the solution: Find the largest number of positive integers in such a way that any two of them $x$ and $y$ ($x\neq y$) ...
2
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2answers
191 views

Fermat's little theorem

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
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1answer
58 views

What the difference between the smallest two numbers from these numbers?

There are infinitely many integers $n$ bigger than $1$, such that if we divide $n$ by any integer $k$ where $2\leq k\leq 11$, the remainder is equal to $1$. What the difference between the smallest ...
6
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1answer
44 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
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2answers
37 views

A confusing probability question..

A magician holds one six-sided die in his left hand and two in his right. What is the probability the number on the dice in his left hand is greater than the sum of the dice in his right?
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3answers
55 views

Probability that an integer is divisible by $8$ [on hold]

If $n$ is an integer from $1$ to $96$ (inclusive), what is the probability that $n(n+1)(n+2)$ is divisible by 8?
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0answers
28 views

How did this GCD formulae came about [duplicate]

A friend of mine was reading a book where a specific case of finding GCD is mentioned. The formulae goes as follows: $gcd(a^m - 1, a^n - 1 ) = a^K-1$ $where, K = gcd(m,n) $ Firstly, I wonder if ...
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0answers
103 views
+50

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + ...
2
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3answers
42 views

Proving that $i! \mid (p-1)\cdot(p-2)\cdots(p-i+1)$ for $i < p$

Started solving this problem: $$ (a+b)^p \equiv a^p+b^p \pmod{p}$$ where $p\in\mathbb{P}$, $a,b\in\mathbb{Z} $ After a few implications I arrived to this $$ i! \mid ...
2
votes
4answers
124 views

An easy way to calculate $12^{101} \bmod 551$?

We learn about encryption methods, and in one of the exercises we need to calculate: $12^{101} \bmod 551$. There an easy way to calculate it? We know that: $M^5=12 \mod 551$ And $M^{505}=M$ ($M\in ...
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votes
3answers
81 views

How to calculate the following limit?

Calculate the following limit where $n \in \mathbb{Z}$ and log is to the base $e$ $$\lim_{x\to\infty} \log \prod_{n=2}^{x} \Bigg(1+\frac{1}{n}\Bigg)^{1/n}$$
2
votes
1answer
60 views

$n^2(n-1)\sigma(n)=0 \mod 12$, where $\sigma(n)$ is the sum of divisors function

I would like to ask about the following question: 1) Clarify and complete a proof (by cases) in which the multiplicative function $\sigma(n)$, this is the sum of divisors function, satisfies ...
3
votes
2answers
117 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$ [duplicate]

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
3
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2answers
39 views

Cantor Sets in perfect sets in the Real numbers

My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets. In one blog a read a proposition. ''Every perfect set ...
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3answers
105 views

Find the smallest natural number $n$

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$. Example of the operation: $123456$ becomes ...
1
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1answer
47 views

The equation $\phi (x)=n$ has only a finite number of solutions.

While reading about Euler's totient function, I came across this question: Prove that for a fixed $n$, the equation $\phi (x)=n$ has only a finite number of solutions. I have thought a lot about ...
4
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5answers
81 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
1
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1answer
46 views

If Wieferich primes are finite…Then what?

I am wondering if $1093$ and $3511$ are the only Wieferich Primes, then what would it imply? (A wieferich prime is a prime satisfying the congruence $2^{p-1}\equiv 1\ mod \ p^2 $). I know of 3 cases; ...
4
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1answer
81 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
0
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1answer
70 views

Are all even numbers the difference of prime powers

Does there exist an even positive integer greater than $100$ (to eliminate trivial cases) that cannot be expressed in the form: $p^2-q$ $p-q^2$ $p^2-q^2$ $p^3-q^3$ where $p$ and $q$ are primes.
0
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3answers
41 views

Finding whole number answers from whole number inputs

How could I find out if the following equation produces a whole number result (y) using only whole number inputs (x). 6y = 2^x
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2answers
89 views
+50

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
1
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1answer
45 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
4
votes
1answer
31 views

What is the maximum value of the LCM of three numbers $\leq n$, as a function of $n$?

Given $n \geq 3$, what maximum LCM of any three numbers $\leq n$ can we obtain? Now, if $n$ is odd, the answer would be $$n(n - 1)(n - 2)$$ because $\newcommand{\lcm}{\operatorname{lcm}}$ ...
3
votes
2answers
63 views

Find the sum$\pmod{1000}$

Find $$1\cdot 2 - 2\cdot 3 + 3\cdot 4 - \cdots + 2015 \cdot 2016 \pmod{1000}$$ I first tried factoring, $$2(1 - 3 + 6 - 10 + \cdots + 2015 \cdot 1008)$$ I know that $\pmod{1000}$ is the last ...