0
votes
1answer
48 views

Abstract algebra

Assuming there is a real number $x$ with $ x^3 =7$, prove that $x$ is irrational. I started the proof by contradiction, and I got to the point that $7q^3 = p^3$, but I don't know what should I do ...
1
vote
1answer
56 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
1
vote
5answers
2k views

Prove that the sum of three consecutive squares, minus two is a multiple of 3

Prove that if you add the squares of three consecutive integer numbers and then subtract two, you always get a multiple of 3.
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 $S$.
1
vote
2answers
98 views

How to show that $a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2} $

I want show the following $$a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2},\ s,\ t\in{\mathbb Q} $$ How can we prove this ? [Add] Someone implies that ...
8
votes
1answer
192 views

How to prove that $\frac{1}{x_1}+\frac{1}{x_2}+…+\frac{1}{x_n}-\frac{1}{x_1x_2…x_n}\in \mathbb{N}\cup \{0\}$

Question: Show that for every natural number $n$ there exist $n$ natural numbers $ x_1 < x_2 < ... < x_n ,$ such that $$ ...
0
votes
1answer
18 views

Formula For Finding the Next Near Consecutive Perfect Square

For any three consecutive members of a sequence, the first and third members are near consecutive. 1 squared is 1. 2 squared is 4. So 1 and 4 are consecutive perfect squares. 1 squared is 1. 3 ...
1
vote
3answers
29 views

How to find the greatest integer, j, such that j * ( j - 1 ) / 2 < k?

How to find the greatest integer, j, such that j * ( j - 1 ) / 2 < k ? Is there a way to find a formula for j in terms of k ? Thanks in advance.
4
votes
2answers
128 views

finding the difference of perfect squares

Find the difference between the smallest perfect square larger than one million and the largest perfect square smaller than one million. I did not want to use a calculator for this question. I ...
1
vote
2answers
55 views

The equation $5^x+2=17^y$ doesn't have solutions in $\mathbb{N}$

Problem: Prove that the equation $5^x+2=17^y$ doesn't have any solutions with $x,y$ in $\mathbb{N}$. I've been analyzing the remainder while dividing by $4$, but I'm getting nowhere.
4
votes
5answers
97 views

Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c$$ Show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but I ...
1
vote
2answers
84 views

Show that there exists no integer b such that f(b) is 1993.

We are given a polynomial $f$ with integer coefficients such that for 4 distinct integers $a_1,a_2,a_3$ and $ a_4$, $f(a_1)=f(a_2)=f(a_3)=f(a_4)=1991$. Show that there exists no integer $b$ such that ...
6
votes
2answers
112 views

Prove that $2^n +1$ in never a perfect cube

Prove that $2^n +1$ in never a perfect cube I've been thinking about this problem, but I don't know how to do it. I know that if $m^3=2^n+1$, then $m$ should be an odd number, but I 'm not able to ...
0
votes
1answer
19 views

Why does the borrowing method for subtracting stop working if the bottom number is bigger?

My brother was given the problem $2.3-4$, and tried to solve it using the standard one over the other format. $.3-.0=.3, 2-4=-2$, answer is $-2.3$. He looks at the answer in the back and sees that it ...
1
vote
1answer
28 views

Finding a nontrivial solutions in natural numbers.

Consider the equation for natural numbers $i,j,k,l:$ $$ (j^2-i^2) (k\cdot l)^2=2\, (l^2-k^2) (i\cdot j)^2. $$ I am trying to prove that it has no solution. To undertand why, let us first consider ...
0
votes
2answers
29 views

replacing numbers to get final anser

I found this question in a random problem solving book that I was reading and wanted to know how you would solve it. I am not sure as how to go about this. Take any positive integer $n$ with fewer ...
0
votes
2answers
35 views

finding values of $x$ in $Z$

Find all values of $x$ such that $\frac{x-4}{2x-3}\in\mathbb Z$? I came up with this question to see if it could be solved based on some other questions I did myself. I thought this could not be ...
1
vote
0answers
23 views

Find the reflection point $P$

On the real number line, paint red all points that correspond to points of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on ...
7
votes
1answer
172 views

How to represent Fermat number $F_n$ as a sum of three squares?

Let $F_n=2^{2^n}+1$ be the Fermat number. How to represent the Fermat number $F_n$ for $n \geq 3$ as a sum of three squares of different natural numbers? For example for $n=3$ we have $$ ...
0
votes
0answers
49 views

Positive integers of sum and products

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$ I just thought about this question and ...
0
votes
2answers
57 views

Pair of positive integers in product sums

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not ...
1
vote
1answer
30 views

Rational solutions to a system of equations

I have a system of equations $$\begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align}$$ Plugging it into a CAS, I see that all the rational solutions to this system have ...
2
votes
6answers
97 views

Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$ [duplicate]

Show $${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$$ A hint is given to consider the expansion $(x-y)^n$ However, when I plug in a number for $n$, I don't get an ...
0
votes
2answers
54 views

Determine the number of digits in $4^n$

Let $n$ be a natural number. How can we determine the number of digits in $4^n$? For example $4^{20}$ has $13$ digits.
3
votes
4answers
108 views

Why is it impossible to find natural numbers $a$ and $b$ such that $12b^2=a^2$?

This was a question in the exercises for an EdX course by Prof Starbird on Effective Thinking through Mathematics which was long over, but I am working through the course at my own pace. I feel that ...
2
votes
1answer
55 views

decomposition into three squares

Doing a coding assignment. And it's basically having a user enter $n$. Then I need to provide (If it exists) $$n = x^2 + y^2 + z^2.$$ Not really sure how to approach this. Any ideas?
3
votes
2answers
84 views

What is the meaning of this Wolfram Alpha result when calculating $3^p = 4^q$?

I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result ...
0
votes
3answers
77 views

Simplifying radical expressions such as $\sqrt{80}$

I am having trouble simplifying a radical expression, such as say...$\sqrt{80}$. What I do is firstly, I do 80/2, then 80/3, then 80/4, then 80/5...etc until I find the largest number that can be ...
5
votes
5answers
195 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
3
votes
3answers
115 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
1
vote
1answer
65 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
0
votes
2answers
51 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
4
votes
1answer
51 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
0
votes
1answer
81 views

Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
2
votes
3answers
117 views

Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
1
vote
2answers
61 views

How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
3
votes
3answers
89 views

searching smallest number that has $40$ distinct positive divisors

What is the smallest natural number such that it has $ 40 $ distinct positive (integer) divisors (inclusive of $ 1 $ and itself? At first I was stunned of seeing the problem.It's not possible to find ...
2
votes
1answer
58 views

Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
3
votes
2answers
59 views

Prove that every non-prime natural number $ > 1$ can be written in the form of $n+(n+2)+(n+4)+…+(n+2m) = p$

I'm trying to prove that every non-prime natural number greater than $1$ can is equal to a sum of consecutive even or odd numbers. This can be resumed in : « $p,m,n \in ℕ$» , «$p > 1$» , «$n > ...
2
votes
2answers
133 views

Counting the number of digits in a concatenation

Concatenate the numbers $2^{1971}$ and $5^{1971}$. How many digits are there in the new number? How do I count them?
3
votes
1answer
113 views

Remainder of $\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$

I am trying to find the remainder of the polynomial division $$\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$$ I know that the answer is 6, but I am not getting that when I ...
2
votes
0answers
116 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
0
votes
0answers
22 views

How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
0
votes
1answer
50 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
1
vote
3answers
63 views

$3$ doesn't divide $x\Longrightarrow\;x^3\equiv\pm1 (\operatorname{mod}9)$ [closed]

I'm stuck in this elementary problem: how can I show that $3$ doesn't divide $x$ implies $\;x^3\equiv\pm1 (\operatorname{mod}9)\:$? Thanks a lot
0
votes
2answers
79 views

Simple math pattern--does it work?

So a friend of mine just pointed this out: $$ \text {If} \ \; 0<a<b \; \text{then} $$ $$ b^3-a^3=(a^2+ab+b^2)(b-a) $$ $$ b^4-a^4=((a^3)+(a^2b)+(ab^2)+(b^3))(b-a) $$ $$ ...
15
votes
2answers
349 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
votes
1answer
58 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
5
votes
1answer
124 views

Prove that $512^3 + 675^3 + 720^3$ is a composite number

We have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite. I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take out some common ...
0
votes
1answer
45 views

prove that $rp+1$ does not divide $p^p-1$

$p$ is a odd prime and $r$ is any odd positive integer. prove that $rp+1$ does not divide $p^p-1$ for any $p$ & $r$. I have expanded $p^p-1$ to $p^p-1=(p-1)(p^{p-1}+p^{p-2}+...+1)$ since ...