Questions on finding integer/rational solutions of polynomial equations.

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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 ...
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1answer
19 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 ...
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1answer
34 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 ...
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2answers
131 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 ...
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2answers
29 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 ...
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1answer
24 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?
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1answer
20 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 ...
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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, ...
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1answer
46 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: ...
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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 $k$ ...
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2answers
71 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} | ...
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2answers
33 views

Is there are integer solutions for this equation: $ 65x-4y= 129$ [on hold]

My question is: Is there are integer solutions for this equation: $$ 65x-4y= 129$$
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1answer
55 views

The diophantine equation $a^7+b^7=7^c$

Determine all the triples of positive integers $a,b,c$ such that $a^7+b^7=7^c$.
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38 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 ...
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1answer
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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 ...
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0answers
34 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.
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1answer
49 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 ...
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1answer
28 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 ...
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1answer
54 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 ...
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2answers
56 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 $\mathbb{Q}$ ...
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0answers
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Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
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1answer
33 views

Solving the Diophantine equation $ax + by = c$ using Maple [closed]

I wrote a program in Maple called EEAsolve (I'm not sure how I can show everybody the code), and what it does is takes 3 parameters from $ax + by = c$: $a$, $b$, and $c$. When I run the program with ...
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2answers
28 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
54 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 ...
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2answers
22 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 ...
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1answer
75 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 ...
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2answers
77 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?
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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 ...
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1answer
37 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 ...
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1answer
29 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.
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1answer
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On no. of solutions of product of positive integers equal to sum [on hold]

$n \ge 2$ be an integer , let $a(n)$ be the no. of solutions in positive integers of $x_1+x_2+...+x_n=x_1x_2...x_n ; x_1 \le x_2 \le ... \le x_n$ , then is it true that $a(n+1)=1 \implies n$ is ...
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1answer
17 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 ...
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64 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 ...
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2answers
52 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
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1answer
61 views

Diophantine Equation: $a^3=a(b^2+c^2+d^2)+2bcd$

Let $a,b,c,d\in \mathbb{Z}$. Solve $$a^3=a(b^2+c^2+d^2)+2bcd$$ I've tried everything but I haven't been able to find a general solution. Note: We may assume $\gcd(a,b,c,d)=1$ because of homogeneity. ...
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2answers
33 views

On the Diophantine Equation $(x-h)^2+(y-k)^2=c$

I am just curious about the equation of the circle centered at (h,k) whose form is we know $(x-h)^2+(y-k)^2=r^2$. If we consider its solution over the set of integers then we have a Diophantine ...
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1answer
51 views

Solving Diophantine equation $1/x^2+1/y^2=1/z^2$

How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation: $$1/x^2+1/y^2=1/z^2$$ Can we conclude that $x$ and $y$ are not coprimes for it to have solution?
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How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
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Primes as sum of squares.

If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$, show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares. It is well ...
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2answers
34 views

Solve $x^3-ax=by$ If $\gcd(x,y)=1$

Solve the diophantine $x^3-ax=by$ If $\gcd(x,y)=1$. Any hint? My first impression is $\gcd(x,b)>1$ and $x^2=ky+a$ for some integer $m$. I conclude that as long as there exists a square integer ...
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30 views

Simultaneous Diophantine Equations

Is there an elegant way to determine whether there is more than one integer solution (not counting plus or minus the same value) in the diophantine equations $x^2 + m=y^2$ $x^2+n=z^2$ where $m$ and ...
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0answers
18 views

Sparse Diophantine Linear Equations System

Do you know any paper/algorithm which deals with Solving the general solution of a sparse Diophantine Linear Equations System?
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119 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)\tag1$$ Finding certain equal sums of like powers that are multi-grades for high powers lead to two questions: If ...
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58 views

Polynomial Diophantine Equation

If $x$,$y$ $\in \mathbb Z$, find all the solutions of $$y^3=x^3+8x^2-6x+8$$ I have tried factorizing the equation but the polynomial on $\text{R.H.S.}$ doesn't have any integral roots. ...
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1answer
43 views

Prove this diophantine equation $b^2=a^3+ac^4$have no integer solution,

show that this diophantine equation: $$b^2=a^3+ac^4$$ has no soluton in non-zero integers [Hint: first show that $a$ must be a perfect square] This problem is from this PDF I know this ...
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An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ was found in 1967 by Lander et al. In 2010, Bremner and Delorme found it had the highly structured form, $$\small(u + 9)^k + (u ...
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34 views

Another look at the trinomial of the form: $ax^n+bx+c=0$

Has the trinomial of the form $ax^n+bx+c=0$ been fully studied for $n>2$? If so, please let me know of any reference or interesting findings. Thanks.
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3answers
56 views

Why are there no integer solutions to $m^2 - 33n + 1 = 0$?

How many solutions does the equation $m^2-33n+1=0$, where $m,n\in\mathbb Z$, have? The answer is no solutions exist. But why?
3
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3answers
77 views

Solving $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$, $\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$

Whether non-zero integers $a, b, c$ with the property that $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$$ and $$\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$$ Calculate ...
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1answer
25 views

Solving $y^{ax}=x^2b$ over integers

Let $x,y,a,b \in \mathbb{Z}>1 $and $\gcd(x,b)=1,$ $y^{ax}=x^2b$, I cannot find any integral solution. What I have done so far: I assume there must be 2 coprime integers $c, d>1$ such that ...