Questions on finding integer/rational solutions of equations.

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1answer
34 views

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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3answers
133 views

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 ...
-2
votes
1answer
81 views

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. ...
-4
votes
1answer
37 views

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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0answers
40 views

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, ...
1
vote
1answer
72 views

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 ...
3
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3answers
67 views

Find $(m,n)$ where $m$ and $n$ are positive integers.

Find all positive integers $m$ and $n$, such that: $$\frac 1m + \frac 1n - \frac 1{mn}=\frac 25$$ Actually, I have already solved this problem using inequality. The solutions I have found are: $$\{(3,...
1
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0answers
12 views

Bound on smallest $n$ for consistency of a system of equations?

Given small $\epsilon>0$ how small should $n\in\Bbb N$ be such that if $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ with $gcd(a,b)=gcd(a,x)=gcd(b,y)=1$ the following relations can hold with constraints $c,d=...
4
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3answers
167 views

How to find integer solutions to $M^2=5N^2+2N+1$?

My number theory is terrible so I don't know what "class" of problem this secretly is. I'm looking for all positive integer solutions to the equation: $M^2=5N^2+2N+1$ That is, I want positive ...
3
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0answers
19 views

Diophantine equation $10^x=yzwt-3$

I have resolved, brute force, the following problem someone asked me: Solve the Diophantine equation $$10^x=yzwt-3\space \text{where}\space \space y,z,w,t \space \text {are distinct primes}$$ $$ *...
1
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1answer
41 views

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: ...
4
votes
1answer
63 views

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 ...
2
votes
1answer
62 views

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 ...
3
votes
2answers
52 views

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-...
2
votes
3answers
97 views

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 ...
0
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2answers
42 views

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 ...
0
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0answers
60 views

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,...
2
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2answers
63 views

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 ...
2
votes
1answer
52 views

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 $k^2$ ...
5
votes
1answer
66 views

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 ...
0
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1answer
34 views

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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2answers
44 views

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, ...
2
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0answers
19 views

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 ...
1
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0answers
21 views

Linear Transformations between solutions to different hyperboloids

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 ...
1
vote
4answers
61 views

Under certain conditions $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}\Rightarrow \{a,b,c\}=\{a',b',c'\}$

Let $a,b,c,a',b',c'\in \mathbb{Z}_{\geq 1}$ be such that $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1,\quad \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}<1. $$ Suppose $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{...
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3answers
72 views

$x^2+y^2=2z^2$, positive integer solutions

Determine all positive integer solutions of the equation $x^2+y^2=2z^2$. First I assume $x \geq y$, and I have $x^2-z^2=z^2-y^2$. Then I have $(x-z)(x+z)=(z-y)(z+y)$, but from here, I don't know how ...
1
vote
2answers
37 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
1
vote
1answer
20 views

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$?

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$? $$p^2 q^2 \geq 3 p^2 q + 3p^2 + 3pq^2 + 3pq + 3p + 3q^2 + 3q + 3$$ I tried to use Wolfram Alpha, and it says that ...
1
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0answers
19 views

Finding the fundamental Pell solution from a system of Pell-like equations

Assume $d$ is a non-square integer, and $r,s,t,w$ are integers, and $n$ and $m $ are integers with $n,m \neq 0,\pm 1$, satisfying the system of Pell-like equations \begin{align} r^2-ds^2 &= m, \\ ...
2
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1answer
83 views

The number of integral solutions $(x,y)$ of $x^3+3x^2y+3xy^2+2y^3=50653$

This was a wonderful question given to me by professor in my last class test. He asked for the solution with the least number of steps. Find the number of integral solutions $(x,y)$ of the ...
2
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3answers
128 views

Integer solutions to $x^2-xy+y^2=1$

What are the integer solutions to $x^2-xy+y^2=1$? (I found the solution below while working on another problem, so I thought I'll add it to the knowledge base here.)
3
votes
2answers
117 views

Find all positive inegers solution for $x^2-xy-y^2=1$

Find all positive inegers solution for the following diophantine equation $$x^2-xy-y^2=1$$ My work so far 1)$$x^2-xy-y^2-1=0$$ $$D=y^2+4(y^2+1)=5y^2+4=k^2, k \in \mathbb Z$$ 2)$$ y^2+xy-x^2+1=0$$...
3
votes
1answer
39 views

Solving $(ap)^2-d(bq)^2=1$ for distinct primes $p,q$

I'm pondering the following claim regarding special cases of the Pell equation. Conjecture: For every pair of distinct primes $p$ and $q$, there exist integers $a$ and $b$, and a non-square integer $...
1
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1answer
21 views

Hilberts tenth problem over $\mathbb R$ with coefficients in $\mathbb Q$

Consider the following decision problem: Given: An equation $f(x_1, \dots, x_n) = 0$ where $f(x_1, \dots, x_n)$ is a polynomial with variables $x_1, \dots, x_n$ and coefficients in $\mathbb Z$. To ...
2
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1answer
76 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; $\gcd(ac+...
3
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1answer
113 views

How many pairs $ (a,b)$ of integers such that , $a^2b^2=4a^5+b^3 $

I would appreciate if somebody could help me with the following problem: $Q$: How many pairs $ (a,b)$ of integers such that $$a^2b^2=4a^5+b^3 $$
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0answers
30 views

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$?

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$? I was wondering about this and was wondering in general are there ways to solve such a question for $f(x,...
0
votes
1answer
71 views

System of equation $x+y+z=2007; xyz=14000$

I have to solve the system of equations $$\begin{cases} x+y+z=2008,\\ xyz=14000, \end{cases}$$ where $x,y,z$ are positive integers such that $1\le x \le y \le z \le 2000.$ My work so far: Let $...
9
votes
3answers
256 views

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact that ...
2
votes
1answer
65 views

Diophantine equation $n^2+n+1=m^3$

Is there an elementary method for solving Diophantine equation $n^2+n+1=m^3$ for integers $m$ and $n$? There is a similar one, which I could solve:$$p^2-p+1=q^3,$$where $p$ and $q$ are prime numbers. ...
5
votes
1answer
114 views

Link between the negative pell equation $x^2-dy^2=-1$ and a certain continued fraction

Consider the generalized continued fraction $$F(x)=(x-1)-\cfrac{(x+1)}{x+\cfrac{(-1)(5)} {3x+\cfrac{(1)(7)}{5x+\cfrac{(3)(9)}{7x+\cfrac{(5)(11)}{9x+\ddots}}}}}$$ I experimentally discovered that at ...
7
votes
2answers
111 views

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$. I think that $x^2 + 2xy + y^2$ and $x^2 + y^2$ are not consecutive squares since ...
4
votes
1answer
59 views

Diophantine equations $x^n-y^n=2016$

Solve equation $$x^n-y^n=2016,$$ where $x,y,n \in \mathbb N$ My work so far: If $n=1$, then $y=k, x=k+2016, k\in \mathbb N$ If $n=2$, then $2016=2^5\cdot 3^2 \cdot 7$ $x-y=1; x+y=2016$ $x-y=...
7
votes
1answer
168 views

Diophantine equation: choosing the right modulus to prove an equation cannot be satisfied

I was looking at this problem, which asks to show that there are no $m,n \in \mathbb Z$ such that $$3n^2+3n+7 = m^3.$$ The result follows immediately from considering the equation modulo $9$ and ...
12
votes
1answer
142 views

$2^n + 3^n = x^p$ has no solutions over the natural numbers

A few weeks ago, I was asked to prove that $2^n + 3^n = x^2$ has no solutions over the positive integers. My proof was: $2^n + 3^n \equiv (-1)^n \equiv \pm 1 \mod{3}\\\text{However, quadratic residue ...
1
vote
1answer
67 views

Preserving modulus residue under division

Modulus residue is preserved or honored (sorry, I don't know the correct term. Is it homomorphism?) under addition and multiplication. For example: 2 + 4 = 6 2 * 4 = 8 Then, making those values ...
7
votes
4answers
132 views

There does not exist any integer $m$ such that $3n^2+3n+7=m^3$

I have this really hard problem that I am working on and I just don't seem to get it. The question is: let $n$ be a positive integer; prove that there does not exist any integer $m$ such that $3n^2+3n+...
4
votes
1answer
77 views

Solving the Diophantine Equation $x^2 - y! = 2001$ and $x^2 - y! = 2016$

I had recently faced a problem: Solve the Diophantine Equation $x^2 - y! = 2001$. Solving it was quite easy. You show how $\forall y \ge 6$, $9|y!$ and since $3$ divides the RHS, it must divide ...
1
vote
3answers
59 views

Integer solutions to $xyz = w^2(x+y+z)$

I'm looking for a way to enumerate all positive integer solutions of the equation $xyz = w^2(x+y+z)$ where $w \le W$ and $1 \le x \le y \le z$. Could anyone provide a hint at how to approach this?
5
votes
3answers
95 views

Proving that the only integer solution of $2x^2+3y^2=z^2$ is $(0,0,0)$

I'd like to prove that the only integer solutions of $$2x^2+3y^2=z^2$$ is $(0,0,0)$. By working in $\mathbb{Z}_2$ and $\mathbb{Z_3}$, I have gone as far as proving that in $\mathbb{Z}$, any integer ...