Questions on finding integer/rational solutions of polynomial equations.

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0
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1answer
48 views

Solve single equation with 2 unknowns?

I don't know how to solve this equation, really tried to Google it but Google foo is weak. $$ \ m^{2} - n^{2} = 1 \\ (m-n)(m+n) = 1 \\ m-n = 1 \quad \& \quad m+n = 1 \\ ? $$ This is about as ...
9
votes
1answer
265 views

How to solve $4x^3-3z^2=y^6$ in positive integers?

Solve in positive integers $$4x^3-3z^2=y^6$$ We are given that $\gcd (x,y) = \gcd (y,z) = \gcd (x,z) = \gcd (x,y,z) = 1$. I do not have the slightest idea how to even begin this question. ...
1
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0answers
30 views

Other Diophantine problems that use a Pell equation

What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like, $$20^2+21^2 = 29^2$$ These are ...
2
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2answers
38 views

Finding integer solutions

Find all integer solutions to the problem $y^2+x^2-6x=0$. How I solved this was to complete the square then finding the coordinates: $(0,0), (6,0), (3,3), (3,-3)$. What I would like to know is there ...
0
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2answers
76 views

Cubes of the Form $3x^2\pm xy+5y^2$, with $x,y$ Coprime

Are there any cubes of the form $3x^2\pm xy+5y^2$, with x, y coprime ? Partly inspired by this question. I tried various computer searches of the form $|x|\le10^a$, $|y|\le10^b$ with $a+b=6$, all ...
0
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0answers
52 views

Solving $x^p+ax^q+b=0$ with $x,a,b$ integer and $p-q>1$

Help solving $x^p+ax^q+b=0$, where $p,q,x \geq 0$ and $a,b \in\mathbb{Z}$. I am well aware of the complexity of this equation. However, I am mostly interested in the following particular case: Given ...
4
votes
3answers
99 views

Natural solutions to $4^n + 2^{n + 1} = 2^{k}$

Is there such an $n$ and $k$ that $$4^n + 2^{n + 1} = 2^{k}$$ with $n, k \in \mathbb N$. I wrote a program and for $n, k < 5000$ have not found a solution. Is this possible?
2
votes
2answers
61 views

Show that the equation $4x=y^2+z^2+1$ has no integer solution

Show that the equation $$4x=y^2+z^2+1$$ has no integer solution. I divided throughout by $4$ to get $$x=\frac{y^2}{4}+\frac{z^2}{4}+\frac{1}{4}$$ but not sure if that is correct
1
vote
1answer
77 views

Solving a diophantine equation

Given the following function: $$f(x) = \sqrt{ (2155 - 6x)^2-4x}$$ where x is an integer and the function also generates an integer value, is there an algorithm to determine its integer solutions?
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2answers
76 views

Is it possible to solve $ax^2+hxy+by^2+c=0$ in integers?

Last time I got stuck in this problem which I have posted earlier. Today I have come accross to this new situation. How to solve the diophantine equation $ax^2+hxy+by^2+c=0$ in integers ? Given all ...
3
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3answers
62 views

How do I prove that this Diophantine equation has no solutions?

How can I prove that $$x^4 - 4y^4 = 2z^2$$ has no solution for positive integers? Thanks.
8
votes
1answer
752 views

Three variable, second-degree symmetric Diophantine equation

Find integers $f,g,h$ such that $3(f^2+g^2+h^2)=14(fg+gh+hf)$. You can do it using a computer or by hand. I tried this problem for ages, got nowhere. Unfortunately I don't know how to program, but I ...
0
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2answers
36 views

Diophantine equation: $3a^2+3b^2+19ab=0$.

Can this equation be solved in integers $a,b$ (Apart from $a=b=0$)? : $3a^2+3b^2+19ab=0$ Thanks!
0
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4answers
46 views

The Diophantine equation $ax+by = b+c$ is solvable in integer $x , y$ iff $ax+by =c$ is solvable.

Let $a,b,c \in \Bbb Z$. Show that the Diophantine equation $ax+by = b+c$ is solvable in integer $x , y$ iff $ax+by =c$ is solvable. We know that a Diophantine equation $ax+by =c$ is solvable iff ...
0
votes
0answers
39 views

Solutions of Pell's equation of the special form

Consider Pell's equation of the form $$x^2-Dy^2=A.$$ I am looking for the reference to the following question: For what values of $D$ and $A$ does it have infinitely many integer solutions of the ...
1
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2answers
32 views

Pythagoras triples

Regarding the parametrization of the pythagora's triples: $x=p^2-q^2$ $y=2pq$ $z=p^2+q^2$ When $x=0, p^2=q^2$. Given that $\gcd(p, q)=1$, is there a contradiction? Why(not)?
0
votes
1answer
45 views

Applying Hensel's lemma

I'm trying to prove that the following equation: $$(x^2 - 2) (x^2 - 17) (x^2 - 2\cdot 17) = 0$$ has solutions $ \pmod{p^k}$ for all $p,k$. It's easy to find nonzero solutions $ \pmod{2,17} $ - and ...
1
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3answers
79 views

Find all integers $x$ such that $x^2+3x+24$ is a perfect square.

Find all integers $x$ such that $x^2+3x+24$ is a perfect square. My attempt: $x^2+3x+24=k^2$ $3(x+8)=(k+x)(k-x)$ Now, do I find solution treating cases? But that doesn't seem very easy. ...
4
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5answers
110 views

Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$.

Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$. My attempt: $(a+1)(a-1)+(b+1)(b-1)=c^2+1$ This form didn't help so I thought of $\mod 3$, but that didn't ...
0
votes
1answer
41 views

Parametrization of solutions of diophantine equation

The issue I discussed in this thread. Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$ Generally speaking at the forum often ask a question about this equation. So I ...
0
votes
1answer
23 views

Parametrising the set of solutions of a simple diophantine equation.

I want to find integers x,y,z, such that k$z^2$ = $x^2$ - $y^2$ for a given integer k. How do I write down the set of solutions? Preferably in parametric form. For a given z, finding all the x and ...
1
vote
1answer
44 views

For which $a>0$ does the equation $x^2+y^2+z^2=a$ have a solution in $\mathbb{Q}_2$?

We want to check for which $a>0$ we have that the equation $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$. $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$ for $x \in \mathbb{Z}_2^{\star}, y ...
13
votes
2answers
189 views

When is $2^n -7$ a perfect square?

This came up while solving another ENT problem. I want to ask when is: $$2^n -7 \text{ where } n\geq 3$$ a perfect square? Specifically, I also wanted to know what would be the solutions when $n$ is ...
0
votes
2answers
33 views

How do I solve $3(2^{x+2}-2^x) = 4a_1a_2a_3$

I encountered this problem but I'm not sure how to solve it since it has 4 unknowns. $$3(2^{x+2}-2^x) = 4a_1a_2a_3$$ What is known is that $x\in\mathbb{Z}$ and $a_1, a_2$ and $a_3$ are digits in a ...
0
votes
1answer
31 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
2
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1answer
38 views

To solve the system of Diophantine equations.

I decided to compile a single task and to record such a system. $$\left\{\begin{aligned}&xt+yw=az^2\\&xw-yt=br^2\end{aligned}\right.$$ $a,b - $ integers that are the problem. It is clear ...
1
vote
3answers
27 views

Where am I going wrong in my linear Diophantine solution?

Let $-2x + -7y = 9$. We find integer solutions $x, y$. These solutions exist iff $\gcd(x, y) \mid 9$. So, $-7 = -2(4) + 1$ then $-2 = 1(-2)$ so the gcd is 1, and $1\mid9$. OK. In other words, ...
2
votes
1answer
51 views

Pythagorean Quadruples Problem

What are all the solutions to $$2^{2x}+2^{2y}+1=n^2 $$ I tried using the parametrization of Pythagorean Quadruples, but it did not work quite well. There are $2$ parametrizations: ...
0
votes
2answers
15 views

Finding $2m+1=2\alpha k+\alpha^2$ quickly

Given some positive integer $m$ I'm looking for all solutions $\alpha,k>0$ to $2m+1=2\alpha k+\alpha^2$ with $0<k^2<2m.$ Right now I'm finding these by looping over each of these possible $k$ ...
1
vote
2answers
77 views

The equation $x^4+y^4=z^2$ has no integer solution

The equation $$x^4+y^4=z^2$$ has no integer solution for $(x, y, z), x \cdot y \neq 0 , z >0$. We suppose that there is a solution $(x, y, z)$. We consider the set $$M=\{z \in \mathbb{N} | ...
-1
votes
1answer
89 views

The diophantine equation $a^7+b^7=7^c$ [closed]

Determine all the triples of positive integers $a,b,c$ such that $a^7+b^7=7^c$.
2
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0answers
59 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
1
vote
1answer
82 views

The diophantine equation $y^2=x^3+7$ has no solutions

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
0
votes
0answers
53 views

Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
3
votes
1answer
58 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
2
votes
1answer
31 views

Solving diophantine equation $6x+9y=1050$ where $x,y \in\mathbb{N}$

I have to solve this Diophantine equation: $6x+9y=1050$, where $x,y \in\mathbb{N}$. I am not sure as to how to solve this for only the whole numbers, but I think I'm doing it right. I used the ...
0
votes
1answer
55 views

Solving the equation $2x^3+3x^2+x-6n^2 = 0$

It came up when I was trying to solve the equality $\sum_{i = 1}^{x}i^2 =n^2$ for integers $x$ and $i$. I've reduced it to the equation $2x^3+3x^2+x-6n^2 = 0$, which I don't know how to tackle. Is ...
5
votes
2answers
62 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
1
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0answers
30 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
2
votes
2answers
33 views

Solve diophantine using modulus

Find all pairs of positive integers $(m, n)$ that satisfy, $mn + 3m - 8n = 59$ Using Modular arithmetic. Okay, this is a diophantine equation, where can I begin?
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0answers
67 views

Form of solutions Pell's equation

I'm studying a proof regarding Pell's equation. It has the form $y^2 - Dx^2 = 1$ with $D \in \mathbb{N}$. Namely that it has an infinite number of solutions if $D$ is not a perfect square. I already ...
1
vote
2answers
25 views

Diophantine equation got wrong

I am trying to understand Diophantine equation article in wiki. They say that in the given equation: $$ax + by = c$$ There will be such integers $x,y$ if and only if $c$ is a multiplier of greatest ...
2
votes
1answer
80 views

How many ordered triples $(a, b, c)$ exist?

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Breaking it up, $500 = 2^2\cdot5^3$ $abc = 2^2 \cdot 5^3 = 2\cdot 2 \cdot 5 \cdot 5 \cdot ...
5
votes
2answers
78 views

How to find all integer solutions of $p^2+q^2=((2q+1)^2+q+1)^2+1$

$$p^2+q^2=((2q+1)^2+q+1)^2+1$$ How do I find integer solutions to this equation? I've already found $(p,q)=(11,1)$. How do I go about finding new ones?
0
votes
0answers
106 views

Diophantine equation 3-rd degree.

When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows. $$b(x^3+y^3)=az^3$$ Where $$b=q^2+3n^2$$ $$a=2(q^2-3n^2)$$ When you can represent the ...
0
votes
1answer
42 views

Solving $y^2 = 1263465 + 144x$ for integers $x,y$

I've thrown this equation up as part of some research I'm doing. $$y^2 = 1263465 + 144x$$ I was hoping there is a quick way to solve this without stepping through all the values. The value I'm ...
0
votes
1answer
30 views

Diophantine equation $^2$

Let $x,y,n \in \mathbb{N} $. The $n$ is given and then we would like to solve: $x^2 + y^2 = n^2$ Is it possible? If yes, how to do it? Thanks in advance.
0
votes
1answer
18 views

Help finding the common factor of $q^{n-1}-p$ and $kq-p$

let $p,q$ be 2 non-zero coprime integers,$n\in\mathbb{Z}>1$ and $k$ any integer. For what $k$ do $q^{n-1}-p$ and $kq-p$ have a common factor? So far, I have been able to come up only with the ...
2
votes
0answers
68 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...
1
vote
2answers
55 views

Diophantine equation with division

How can I find all the cases where y is positive integer in the next equation: $$\frac{ax + b}{c-x} = y$$ $a,b,c,x$ are not negative integers $a,b,x < c$ $ax + b = 0$ is a trivial solution