-7
votes
0answers
42 views

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7)

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7 I've tried rearranging each equation to isolate for one variable ex: z^2-xy=-7 --> z= x^2-18/y after, I ...
4
votes
1answer
48 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
2answers
34 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
14
votes
2answers
231 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 ...
0
votes
1answer
27 views

Base convertion and equations

I am studying for an exam in my course, and I will certainly have a question of the kind: In what base is the equation right, for example: 42-3=36 Another ...
3
votes
1answer
57 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?
1
vote
1answer
26 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
3
votes
1answer
46 views

Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $ pq = r + 1 $ $ 2(p^2+q^2) = r^2 + 1 $ I do not know how to start looking for an answer.
8
votes
4answers
377 views

Equation with an infinite number of solutions

I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions? 2. Which are the solutions over the integers?($ x=3 $ and $ y=3 $ is one) ...
7
votes
4answers
100 views

Diophantine equation: $(x-y)^2=x+y$

I have to solve the following equation: $(x-y)^2=x+y$, where $x$ and $y$ are non-negative integers. This equation has an infinite number of solutions, but how to prove that there exists a positive ...
1
vote
2answers
72 views

Can 11 be represented by $a^2 - 3b^2$ where a and b are integers?

I know the answer is no, just wan't to know how. From a similar question on the site I got that $a^2 - 3b^2$ should always equal a square modulo 3 which 11 is not. But I don't understand how to get to ...
0
votes
1answer
103 views

How to prove that $x^2 - 3y^2 = 11$ is not possible as long as x and y are integers.

Really No idea how I can go about solving it. What I did was, $$y = \sqrt{\frac{x^2 - 11}{3}}$$ but cant go beyond that. Can anybody help?
4
votes
3answers
262 views

Proof that the equation $x^2 - 3y^2 = 1$ has infinite solutions for $x$ and $y$ being integers

I have seen the pell's equation wiki page but I need to prove this from scratch without mentioning any formula. I have also seen multiple answers on this site but the answers tend to skip over and ...
0
votes
2answers
95 views

Proof that $x^2 -3y^2 = 1$ has infinite solutions. ( x and y are integers)

I have to explain this to my brother who is in eighth grade and I would really love if you could tell this in simple terms (I'm no Maths guy).
3
votes
2answers
179 views

A partition of tenth powers into ten parts with equal sums

Is there a natural number n for which the numbers $1^{10},2^{10},3^{10}\ldots, n^{10}$ we can put into 10 groups, such that the sum of the numbers in each group is the same?
2
votes
1answer
152 views

Solve in $\mathbb Z$ the equation: $x^5 +15xy + y^5=1$

Solve in $\mathbb Z$ the equation: $x^5 +15xy + y^5=1$ I tried: $x(15y+x^4)+y^5=1$ But don't have much ideas on how to continue, thanks!
5
votes
2answers
115 views

Sequential sums $1+2+\cdots+N$ that are squares [duplicate]

While playing with sums $S_n = 1+\cdots+n$ of integers, I have just come across some "mathematical magic" I have no explanation and no proof for. Maybe you can give me some comments on this: I had ...
6
votes
3answers
128 views

Find all real solutions for $16^{x^2 + y} + 16^{x + y^2} = 1$

Find all $x, y \in \mathbb{R}$ such that: $$16^{x^2 + y} + 16^{x + y^2} = 1$$ The first obvious approach was to take the log base $16$ of both sides: $$\log_{16}(16^{x^2 + y} + 16^{x + y^2}) = 0$$ ...
8
votes
3answers
448 views

Integral solution of the equation $x^2+y^2+z^2 = 2xyz$

Calculation of all Integral solution of the equation $x^2+y^2+z^2 = 2xyz$ $\bf{My\; Try}::$ Let we will calculate for $x,y,z>0$. Then Using $\bf{A.M\geq G.M}$ $x^2+y^2\geq 2xy$ sililarly ...
0
votes
1answer
55 views

solution of $y^2 - x = 15$ and $x^2 -xy = 2009$

Find all the integer solutions to the equations: \begin{eqnarray} y^2 - x &=& 15 \\ x^2 -xy &=& 2009 \end{eqnarray} Not sure how to solve this :/, tried the usual algebra way ...
1
vote
1answer
65 views

Find the general solution and positive integral solutions of 775x -711y =1

Problem : Find the general solution and positive integral solutions of 775x -711y =1 My approach : For the equation ax -by =c , the general solution can be given as : Let h,k be a solution of ...
0
votes
1answer
210 views

How to Solve an equation with mod for a variable?

I have following equation to be solved, but I am having some trouble in making an understanding and doing so. (d * e) % v = 1 e and v are known. How to solve this ...
0
votes
1answer
31 views

How to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants?

Is there a general formula to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants? Because I was struggling trying to find solutions to this ...
0
votes
3answers
420 views

Required solution to an aptitude question

The following is an aptitude problem (question no: 29-32), I am trying to solve:- Questions 29 - 32: A, B, C, D, E and F are six positive integers such that B + C + D + E = 4A C + ...
3
votes
0answers
78 views

Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
3
votes
4answers
119 views

Find $a,b\in\mathbb{Z}^{+}$ such that $\large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6}$

find positive intergers $a,b$ such that $\large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6}$ Here i tried plugging $x^3=a,y^3=b$ $(x+y-1)^2=x^2+y^2+1+2(xy-x-y)=49+20\sqrt[3]{6} $ the right ...
6
votes
2answers
175 views

Prove that $(3+5\sqrt{2})^m=(5+3 \sqrt{2})^n$ has no positive integer solutions?

Is my proof ok? I set $b=3+5 \sqrt{2}$, so that we have $b^m=(b+2-3 \sqrt{2})^n$ , or $b^m=(b+\sqrt2(\sqrt2- 3))^n$. Since $RHS<LHS$, $n>m$ . However, from what we know about binomial ...
6
votes
1answer
87 views

System of three equations in three variables?

Fibonacci apparently found some solutions to this problem: Find rational solutions of: $$x+y+z+x^2=u^2$$ $$x+y+z+x^2+y^2=v^2$$ $$x+y+z+x^2+y^2+z^2=w^2$$ How would you find solutions to this using ...
5
votes
6answers
320 views

Is my proof correct? $2^n=x^2+23$ has an infinite number of (integer) solutions.

This is how I tried to prove it. Is it correct? Thanks!! $2^n = x^2+23$ $x^2$ must be odd, therefore $x^2 = 4k+1$, where $k \in \mathbb{N}$. $2^n=4k+24$ $k=2(2^{n-3}-3)$ Since $x^2=4k+1$,$ \ \ \ ...
0
votes
3answers
141 views

Finding the number of integer solutions, why is this wrong?

The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
1
vote
1answer
118 views

A Diophantine equation and decimal digits

Solutions of the Diophantine equation $a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$ are 12=3*4, 56=7*8, 67100672=8191*8192. Are there more solutions/examples like that or a generalization of the ...
35
votes
6answers
1k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
3
votes
4answers
71 views

Solve : $\frac{n}{2}(n+1)=2014+2k$.

$n,k$ are positive integers and $n>k$, solve the equation : $$\frac{n(n+1)}{2}=2014+2k.$$ the first thing I did is to write the LHS as $(2n+1)^2$ but I face an equation like $ak+b=m^2$, I know ...
1
vote
1answer
122 views

Alternative solutions to $n^2+n = k^2+k + 2kn$

Consider this equation: $n^2+n = k^2+k + 2kn$ I want to find the set of non-negative integer n,k that satisfies the equation. I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
0
votes
3answers
65 views

Solving the algebraic equation

I am trying to solve this: $$x-40={-400\over x}$$ The answer must be $x=20$ Please give step by step explanation.
6
votes
3answers
262 views

How many solutions are possible to this equation?

Given $$A+2B+3C=N $$ where $N$ is a given positive integer. $A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$. How many solutions will be there to this equation?
1
vote
1answer
843 views

Solving a quadratic diophantine equation in two variables

I have an equation in the following form: $$6mn+m+n=x$$ $$m,n,x\in\Bbb Z; \qquad0 < m,n$$ If I were given a value for $x$, how would I go about finding solutions to this equality for $a$ and $b$ ...
1
vote
3answers
338 views

Finding the sum of all solutions

$2x + 3y = n$ has exactly $2011$ non-negative integral solutions. Determine the SUM of the possible values of $n$.
9
votes
2answers
191 views

Determining the number $N$

Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
1
vote
3answers
163 views

Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$

Note: this is a programming challenge at this site For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$ find the number of positive integral solutions for ...
6
votes
3answers
416 views

The positive integer solutions for $2^a+3^b=5^c$

What are the positive integer solutions to the equation $$2^a + 3^b = 5^c$$ Of course $(1,\space 1, \space 1)$ is a solution.
8
votes
3answers
196 views

$a+b=c \times d$ and $a\times b = c + d$

There is a 'nice' relationship between the integers (1,5) and (2,3) as $$1+5=2 \times 3;$$ $$1\times 5 = 2 + 3.$$ So I tried to find all positive integers pairs $(a, b)$ and $(c, d)$ such that ...
7
votes
2answers
274 views

Find the possible values of $a$, $b$ and $c$?

Given $(a,\space b,\space c)\in \mathbb Z^3$ and that $$\sqrt[3]{\sqrt{a}+\sqrt{b}} + \sqrt[3]{\sqrt{a}-\sqrt{b}} = c$$ Find the possible values of $a $, $b$, and $c$.
1
vote
2answers
107 views

number of integral solutions for $x^2+y^2=5^k$

Prove that the equation $x^2+y^2=5^k$ has $4k+4$ integral solution. Any ideas would be appreciated. Thanks
0
votes
3answers
306 views

Are there any integer solutions to $a^3=b^2$?

I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$.
4
votes
0answers
158 views

How many natural numbers $x, y$ are possible if $(x - y)^2 = \frac{4xy}{(x + y - 1)}$.

How many natural numbers $x$, $y$ are possible if $(x - y)^2 = \frac{4xy}{x + y - 1}$. Does this system has infinite solutions which can be generalized for some integer $k \geq 2?$ $(x - y)^2(x + y) ...
2
votes
2answers
92 views

Unique solutions for $ab = n ^ 2$

How many unique solutions are there to the equation $ab = n^2$ , where $n$ is a constant, $a,b \geq 1$ and $a,b,n$ are integers. Is there any way of counting the number of solutions?
7
votes
3answers
584 views

Sum of three consecutive cubes

When I noticed that $3^3+4^3+5^3=6^3$, I wondered if there are any other times where $(a-1)^3+a^3+(a+1)^3$ equals another cube. That expression simplifies to $3a(a^2+2)$ and I'm still trying to find ...
2
votes
1answer
228 views

Number of integer solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$

What is the number of integer solutions of: $$\frac{1}{x} + \frac{1}{y} = \frac{1}{1000}$$ How to solve these type of problems if am comfortable of solving $x+y=z$. But how to do if multiplicative ...
0
votes
1answer
372 views

The diophantine equation $a+b+c+d+e = abcde$

Now you may think that I am annoying, but if I am not asking this question, then it seems not so complete and I can't grasp the whole idea... refer to this question: Positive rationals satisfying ...