# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### Power Diophantine equation: $(a+1)^n=a^{n+2}+(2a+1)^{n-1}$

How to solve following power Diophantine equation in positive integers with $n>1$:$$(a+1)^n=a^{n+2}+(2a+1)^{n-1}$$
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### Sum of two consecutive squares equal square

$N^2 + (N+1)^2 = K^2$, find all solutions for $N<200$ I have done some factoring and also realized that $K=[n\sqrt{2}]+1$ in eventual solutions, where $[x]$ denotes the greatest integer less than....
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### solve pairs of two variable simultaneous linear modular equations

I’m looking for a method to solve pairs of simultaneous linear modular equations, such as 323x + 37y = 0 Mod 243; -397x + 683y = 0 Mod 32 I’ve simplified this to 80x+37y = 243g; 19x+11y = ...
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### Odd binomial sum equality has only trivial solution?

Suppose $$\sum_{k\ {\rm odd}}^n {n \choose k} 2^{(k-1)/2} = \sum_{k\ {\rm odd}}^m {m \choose k} 2^{(k-1)/2} 3^{(m-k)/2}.$$ Does $m=n=1$? Clearly $m \leq n$, and for every $n$ there is at most one $m$....
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### the number of integer solutions to $y^p = x^2 +4$

Let $p>2$ be prime, investigate the number of integer solutions to $$y^p = x^2 +4$$. The first part of the question was find solutions to the equation $y^3 = x^2 +4$, I could do this and I see the ...
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### Question about $F(x,y)=m$

Let $F(x,y)$ be a homogeneous polynomial of degree $\ge3$ with mutually prime coefficients, then we consider the problem $$F(x,y)=m\tag1$$ such that $m$ is an integer, we set $f(x):=F(x,1)$ then why ...
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### Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$

I started like this : $a^2+c^2=b^2(a^2-1)\\c^2 +1=(a^2-1)(b^2-1)$ but it's leads to nowhere. can you help please ?
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### number of integer solutions to $2x_1 + x_2 + x_3 = n$

I'm working on a problem for which I need to efficiently compute the number of integer solutions to equations of the form $x_1 + \cdots + x_k = n$ with some subset of $\{x_1, \dots, x_n\}$ equivalent. ...
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### A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
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### Find all integral solutions of the equation $x^n+y^n+z^n=2016$

Find all integral solutions of equation $$x^n+y^n+z^n=2016,$$ where $x,y,z,n -$ integers and $n\ge 2$ My work so far: 1) $n=2$ $$x^2+y^2+z^2=2016$$ I used wolframalpha n=2 and I received the ...
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### Prove an equation has no integer solutions… [closed]

I know that ${x^3} - 8{y^3} = 12$ has no integer solutions but how can I prove it? If I had to sit down with someone and convince them (at least, fairly) rigorously that it has no integer solutions. ...
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### No solution in naturals

Prove that there is no pair $(x,y)$ of positive integers such that $$axy-b=x(x-c)+y(y-d)$$ where $a,b,c,d$ are positive integers such that $a>b>(\frac{1}{2} \cdot max\{c,d\})^2$.
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### Is there a short proof for Bézout's lemma when $\gcd(a,b)=1$?

Bézout's lemma states that there exist integers $x$ and $y$ such that $$ax+by=\gcd(a,b)$$ Is there some short proof for this when $a$ and $b$ are coprime? As opposed to something like this, like, ...
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### Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions.

Show that $x^3 + y^3 + z^3 + t^3 = 1999$ has infinitely many integer solutions. I have not been able to find a single solution to this equation. With some trial I think there does not exist a ...
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### Trying to understand Diophantine equations.

I'm revising for an upcoming exam and do not understand how you solve these questions at all. It all seems like a bit of trial and error to get any type of solution. The questions I am looking at are: ...
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### Diophantine equation involving factorials

$$x!+y=y^3$$ $$y=\sqrt[3]{x!+\sqrt[3]{x!+\sqrt[3]{x!+\cdots}}}$$ The only integer solutions to these identities that I have found are: $$3!+2=2^3$$ $$4!+3=3^3$$ $$5!+5=5^3$$ $$6!+9=9^3$$ I ...
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### Diophantine $4x^3-y^2=3$

I am interested in how to tackle this Diophantine equation: $$4x^3-y^2=3$$ The solutions I have found so far are $(1,1)$ and $(7,37)$. Are there any more? I have looked up various material on cubic ...
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### Diophantine $xy+yz+zx=4(x+y+z)$

How do you solve the Diophantine equation $xy+yz+zx=4(x+y+z)$ for positive integers $x,y,z$? My approach was to consider $d=\gcd(x,y,z)$. I could just about show that the equation has no positive-...
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### If $ab^2+1 = c^2+d^2$ with $a$ squarefree, what [else] can be said about $a$?

What is known about squarefree integers $a$ where there exist non-zero integers $b$, $c$, and $d$, with $\gcd(b,c)=\gcd(b,d)=1$, such that $$ab^2+1=c^2+d^2$$ ? EDIT: As pointed out by individ, if an ...
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### How to find all Possible values for A and B, given only one equation? [closed]

Given that : $$\frac{1}{a} + \frac{1}{b} = \frac{1}{12}$$ With a & b integers How can I find all possible values of a and b with only one equation (this one ?) . From what I'v learned in ...
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### Number of solutions to a modular equation of a specific form

I struggle with this Exercise, or at least the part where one should prove how many solutions there are. Simply inserting f=0 contradicts the suggested number of solutions. Let $p$ be an odd prime,...
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### Find triples $(a,b,c)$ of positive integers such that…

Find the triples $(a,b,c)$ of positive integers that satisfy $$\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=3.$$ I found this on a local question paper, and I am ...
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### Find all integer values of $x$ such that $x^2 + 13x + 3$ is a perfect integer square.

Question: Find all integer values of $x$ such that $x^2 + 13x + 3$ is a perfect integer square. What I have attempted; For $x^2 + 13x + 3$ to be a perfect integer square let it equal $k^2$ ...
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### Diophantine Equation $x^2+y^2+z^2=c$

$x^2+y^2+z^2=c$ Find the smallest integer $c$ that gives this equation one solution in natural numbers. Find the smallest integer $c$ that gives this equation two distinct solutions in ...
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### Smallest number of coins to guarantee exact change?

What is the smallest number of coins (excluding 50 cent piece) thats value can sum to any amount .01 to .99? This is a question that I came up with today and my immediate thought is 3Q, 2D, 1N, ...
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### Solve Linear Diophantine $12x+18y = 54$

What is asked? As the title suggests I'm trying to solve a very simple Linear Diophantine Equation: $$12x + 18y = 54$$ Also find an expression for all integer solutions What have I done? Firstly, ...
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### If $a\not\equiv 0\mod{p}$ then there are $p-1$ solutions (ordered pairs) to $x^2-y^2\equiv a\mod{p}$

Let $p$ be an odd prime, and let $a\in\mathbb{Z}_p$ such that $a\not\equiv 0$. I need to show that there are $p-1$ ordered pairs $(x,y)$ such that $x^2-y^2\equiv a \mod{p}$. As I see it, the ...
Is there a way to develop a linear transformation which will always send solutions of one hyperboloid to another? (for example the hyperboloids: $$a^2+b^2-c^2=4$$ and $$d^2+e^2-f^2=9$$ )I know that ...