# Tagged Questions

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 ...
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 ...
34 views

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 ...
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 ...
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 $$$$\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}$$$$ ...
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 ...
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 ...
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.
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?
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 $b$ ...
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 $n$.
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 ...
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 ...
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.
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 ...
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 $c$.
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
### Are there any integer solutions to $a^3=b^2$?
I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$.