Questions on finding integer/rational solutions of polynomial equations.

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0
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1answer
147 views

Finding the largest 3-digit number $\; \overline{abc}\;$ s.t $\; \overline{abc}=100a+10b+c \equiv a+b^2+c^3$

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
3
votes
1answer
84 views

Number Theory : Solving $x^2$ $+$ $y^2$ $=$ $2^{10}$ - $1$

I was working my way through some basic number theory problems and was all thumbs while solving this problem : List all the pairs of integer solutions $(x, y)$ of the Diophantine equation : $x^2$ ...
1
vote
1answer
44 views

Solution to integers of the form $x^n\:+y^n\:=z^n\:+w^n$.

I am wondering if integers of the form $x^n\:+y^n\:=z^n\:+w^n$ have a solution if none of $x,\:y,\:z,\:or\:w$ are equal.
0
votes
0answers
27 views

Minimize multivariate (multivariable) polynomial over the integers

I'd like to minimize the following polynomial in 6 variables $h_0,h_1,g_0,g_1,g_2,g_3$: $$ g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - ...
1
vote
2answers
57 views

A system of quadratic Diophantine equations with four variables

Is the following system has any positive integer solution $(x,y,u,v)$? $$\begin{cases} x^2+y^2=u^2\\ x^2-y^2=v^2 \end{cases}$$ I can prove that any pair of these integers can be relatively prime, but ...
0
votes
1answer
27 views

Impossibility of Equation

Prove that there are no solutions to $ k^2 = x^4 + 2x^3 + 2x^2 + 2x + 1 $ in $ \mathbb Z^+$. I have tried a bounding argument so far, placing $k^2$ in between $x^4$ and $(x+1)^4$, but I am unable to ...
1
vote
3answers
91 views

Why does $5x^2+6xy+2y^2=2yz+z^2$ have no integer solutions?

Why does $5x^2+6xy+2y^2=2yz+z^2$ have no primitive integer solutions? Modulo $2$ says that $x$ and $z$ are odd. Modulo $3$ says that $x=0 \bmod 3$ and $y=-z \bmod 3$. I cannot get anything modulo $5$. ...
3
votes
0answers
66 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...
1
vote
0answers
36 views

How to solve $xy+ax+by+c=0$ in inetegrs?

Respected all. Before I ask your support, let me show you what I have done and have got stuck. We are willing to solve $2x+3xy+4y=5$. So this is what I have done. The given equation becomes ...
0
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0answers
27 views

Mixed real/integer linear system of diophantine equation

Are there papers for mixed real/integer linear diophantine system of equation? So where not all variables are needed to be integer, so some can be real too. ...
0
votes
1answer
70 views

Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y. we know that (in this paper): ...
1
vote
1answer
50 views

Thue's Theorem of Diophantine approximation

I'm trying to read the book Lecture Notes on Diophantine Analysis by Zannier and he says that the following theorems are equivalent: Theorem 1. Let $\xi$ be an algebraic real number of degree ...
1
vote
1answer
69 views

solving linear diophantine equation with inequalities

Is there an easy way to solve linear Diophantine equations with inequalities? For example, say I have $a_1x_1 + a_2x_2 \equiv k \mod m$ where: $a_1, a_2, k, m$ are given I already know $b_1, b_2$ ...
2
votes
1answer
78 views

Integer solutions to $ n^2 + 1 = 2 \times 5^m$

What are the integer solutions to the diophantine equation $$n^2 + 1 = 2 \times 5 ^m? $$ We have $(n,m) = (3,1), (7, 2) $ as solutions. Are there any more? This seems like it would be a well ...
4
votes
0answers
152 views

Diophantine equation: $7^x=3^y-2$

I've tried using mods but nothing is working on this one: solve in positive integers $x,y$ the diophantine equation $7^x=3^y-2$.
1
vote
0answers
40 views

four different integers exist, what is the least product value? [duplicate]

Four different positive integers $a, b, c, d$ are such that $a^2 + b^2 = c^2 + d^2$. What is the smallest possible value of $abcd$? $$a^2 - c^2 = d^2 - b^2$$ $$(a-c)(a+c) = (d-b)(d+b)$$ ...
3
votes
0answers
81 views

Show that $a^2 + b^2 = c^3 $ has infinitely many solutions [duplicate]

Show that $a^2 + b^2 = c^3 $ has infinitely many solutions in $ \{ (a,b,c) \in \Bbb Z ^3 | (a,b)=1, (a,c)=1, (b,c)=1 \}$ . Describe all these solutions. I don't know how to approach this question. ...
3
votes
0answers
56 views

Pythagorean rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons ...
0
votes
1answer
40 views

The system of Diophantine equations.

Often seen similar systems of equations. Usually consider such systems in which decisions no. Such as there. Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$? I think it would be ...
0
votes
1answer
50 views

Number of solutions of $x+y=2n$ and of $x+y+2z=n$

Could anyone explain how to solve this problem please. Let $n$ be an odd integer and $$n\ge5$$ find the number of pairs $(x,y)$ of positive integer which satisfy the equation $$ x+2y=n$$ find ...
6
votes
2answers
135 views

Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$? [duplicate]

Question: Are there $a,b,c,d \in \mathbb N$ such that $$a^2 + b^2 = c^2 \ \ \text{and} \ \ b^2 + c^2 = d^2$$ I'm a bit lost here.
3
votes
3answers
133 views

Do four natural numbers exist which satisfy these constraints?

Do four natural numbers $a,b,c$ and $d$ exist such that the following three conditions are true? $$a^2+b^2+2d^2=c^2$$ $$\sqrt{a^2+d^2}\in\mathbb{N}$$ $$\sqrt{b^2+d^2}\in \mathbb{N}$$
3
votes
2answers
145 views

Another method of proving $x^3+y^3=z^3$ has no integral solutions?

The equation $$ax^2+by^2=z^3$$ has the following parametrization: $$y=q(3ap^2-bq^2)$$ $$x=p(ap^2-3bq^2)$$ $$z=ap^2+bq^2$$ can we deduct from that the Diophantine equation $$x^3+y^3=z^3$$ has no ...
0
votes
1answer
28 views

$q \equiv 2 (\mod 3)$ be a prime , then does there exist non-zero integers $a,b,c$ such that $a^2+ab+b^2=qc^2$ ?

Let $q$ be a prime such that $q \equiv 2 (\mod 3)$ , then is it true that $a^2+ab+b^2=qc^2$ has no solution in non-zero integers $a,b,c$ ?
0
votes
0answers
28 views

Binary quadratic forms

Is it known exactly for which integers $a,b,c$ the equation $ax^2+bxy+cy^2=0$ has a nontrivial solution? If not, what is known about this problem in general? (I am asking about the specific problem of ...
0
votes
1answer
29 views

Question about negative Pell's Equation

Is it true that, if $a^2-Db^2=-1$ is solvable in integers, then so is $x^2-Dy^2=D$ (*)? For $D=5$ this is true, you can take $x=5$ and $y=2$, and indeed $5^2-5(2^2)=5$, so (*) is solvable. Is this ...
1
vote
1answer
36 views

Question about Negative Pell Equations

Does every soluble negative pell equation, $a^2-Db^2=-1$, have infinitely many integer solutions $(a,b)$ where $a,b$ are both positive integers?
3
votes
1answer
76 views

Pell Equations: $a^2+4=5b^2$

This is a challenge problem in the Pell Equations chapter of my number theory book, but I'm not seeing the connection to Pell Equations. The Pell Equation with the coefficient $5$ is $5b^2+1=a^2$, but ...
6
votes
3answers
172 views

Linear Diophantine Equations in Three Variables

$$ 3x+6y+5z=7 $$ The general solution to this linear Diophantine equation is as described here (Page 7-8) is: $$ x = 5k+2l+14 $$ $$ y = -l $$ $$ z = -7-k $$ $$ k,l \in \mathbb{Z} $$ If I plug the ...
3
votes
3answers
70 views

Getting integer solutions for equation $x^{2}-y^{4}=336$

I need to get integer solutions for the next equation: $$x^{2}-y^{4}=336$$ I know equations that look like $x^{2}-y^{2}=n$ have solutions $x$ and $y$ where $x=\frac{a+b}{2}$ and $y=\frac{a-b}{2}$, $a$ ...
0
votes
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
266 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
vote
0answers
34 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
votes
2answers
44 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
votes
2answers
83 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
votes
0answers
53 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?
0
votes
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
votes
3answers
68 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
767 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
votes
2answers
39 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
votes
4answers
49 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
45 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
vote
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
48 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
vote
3answers
82 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
votes
5answers
120 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
56 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 ...