Questions on finding integer/rational solutions of polynomial equations.

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2
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0answers
34 views

Finding solutions to a symmetric divisibility condition $x\mid p(y),\;y\mid p(x)$

In general, are there strategies for finding all integers $x$ and $y$ such that $x \mid p(y)$ and $y \mid p(x)$ for some polynomial $p$ with integer coefficients? For example, could we find all ...
1
vote
1answer
36 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and ...
0
votes
2answers
43 views

Arithmetic sequence of exponentials

There is an arithmetic sequence $2^a, 3^b, 4^c$ such that a,b and c are positive integers. The question posed is to find ALL possible ordered triplets $(a,b,c)$. The constant difference d between ...
2
votes
1answer
59 views

Prove (or provide a counterexample): no pair of primitive Pythagorean triples (a,b,c) and (2a,k,c) exists.

A primitive Pythagorean triple is an ordered set of coprime integers (a,b,c) such that $a^2+b^2=c^2$. Show that the system of Diophantine equations $$a^2+b^2=c^2$$ $$4a^2+k^2=c^2$$ have no solutions.
6
votes
2answers
133 views

Is $y^2=x^3+7$ unsolvable modulo some $n$?

The equation $y^2=4x^3+7$ has no integral solution since $y^2\equiv4x^3+7\pmod4$ has no solution (i.e. has no solution in $\Bbb{Z}/4\Bbb{Z}$). It is well known that $y^2=x^3+7$ has no integral ...
3
votes
1answer
110 views

When is $c^4-72b^2c^2+320b^3c-432b^4$ a positive square?

In trying to solve a certain [third-degree] Diophantine equation, I have used the quadratic equation to determine that $$c^4-72b^2c^2+320b^3c-432b^4$$ must be a positive integer square, where $c$ and ...
3
votes
0answers
52 views

About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
1
vote
2answers
30 views

Quadratic Diophantine Problem in two variables

I have a quick question in regards to solving a quadratic two-variable diophantine problem. The equation is $6x^2 - 2xy + 3y - 17x = 6$. My attempt thus far starts by making y the subject: $$y = ...
0
votes
0answers
34 views

What is known about the Heronian primes?

A Diophantine equation $$x^3 - Dy^3 = 1$$ always has a trivial solution $x = y^3 + 1$. It appears that a non-trivial (that is those with $x$ smaller than trivial) solution exists iff $y$ is a Heronian ...
1
vote
1answer
53 views

Find all natural solutions to $x^2+2y^2 = z^2$ [duplicate]

I need to find all natural solutions to $x^2 + 2y^2 = z^2$ What I tried: I did $\pmod 2$ to the equation receiving $z^2 - x^2 \equiv 0 \pmod 2$. Then there are two possibilities: $x^2 \equiv 0 ...
2
votes
6answers
67 views

find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number ...
0
votes
1answer
39 views

Postive integer solution to this equation $a^2+b^2+c^2+1=kabc$

Frobenius and Hurwitz( in 1880) prove this theorem: For any positive integer $k$ other than 1 or 3, the equation $a^2+b^2+c^2=kabc$ has no integral solution except (0,0,0). My Question,How to solve ...
4
votes
2answers
132 views

Diophantine Equation $ x^n + y^n =z^n (x<y, n>2) $

I am looking for simple college level algebraic solution to prove that $x$ and $y$ ($x$ < $y$) for the above equation can't be prime numbers. (I know more complex and involved solution using high ...
1
vote
1answer
82 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation ...
2
votes
3answers
86 views

Almost extended Euclidean algorithm - $ax+by=\gcd(a,b)+2$

So I have this equation: $$\eta+2=2g+1n,$$ where $g,n \in \mathbb{N}_{\geq 0}$ and $\eta \in \mathbb{N}_{>0}$. I want to find all possible integer-valued 2-tuples $(g,n)$ that satisfy this ...
1
vote
1answer
44 views

When do three cubics form an arithmetic progression?

Are there any solutions to the diophantine equation $x^3+y^3=2z^3$ other than the trivial ones? What about $x^4+y^4=2z^4$? I think I remember these equations in one of Euler's work, but having ...
0
votes
0answers
52 views

Solving a diophantine equation in an elementary way

I was trying to solve $x^2+1=y^3$ and found this answer: Does an elementary solution exist to $x^2+1=y^3$? but I'm having trouble understanding it. In the second last paragraph, why is there a ...
1
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1answer
50 views

How do I prove this claim?

Claim :Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation $ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m $ has a positive integer solution $(x, y) \neq (1, 1)$ if and ...
2
votes
3answers
68 views

Conjecture about linear diophantine equations

I've been dabbling with linear Diophantine equations and came across a rather interesting pattern that I would like to conjecture as true but I have no idea how about to come up with a proof. Let ...
4
votes
1answer
46 views

Solving a Diophantine equation with LTE

Show that only positive integer value of $a$ for which $$4(a^n+1)$$ is a perfect cube for all positive integers $n$, is $1$. Rewriting the equation we obtain: $$4(a^n+1)=k^3$$ It is obvious that $k$ ...
0
votes
0answers
29 views

For which values of $\theta $ does this claim true?

According to the conjecture of , L. J. Lander, T. R. Parkin, and John Selfridge (1967):I suppose this claim : claim :let : $$\sum_{i=1}^n a^{\cos\theta}_i =\sum_{j=1}^m b^{\cos\theta}_j ,$$ for ...
7
votes
4answers
143 views

Find the number of positive integers solutions of the equation $3x+2y=37$

Find the number of positive integers solutions of the equation $3x+2y=37$ where $x>0,y>0,\ \ x,y\in \mathbb{Z}$ . By trial and error I found $$\begin{array}{|c|c|} \hline x & y \\ ...
3
votes
3answers
112 views

Find all Integral solutions to $x+y+z=3$, $x^3+y^3+z^3=3$.

Suppose that $x^3+y^3+z^3=3$ and $x+y+z=3$. What are all integral solutions of this equation? I can only find $x=y=z=1$.
5
votes
1answer
62 views

How to extract solutions to a Pell's equation satisfying certain congruences?

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
-1
votes
1answer
78 views

How we can deal with this equation $a^{n}+b^n=c^{n}$ if it was gaven to have solutions in primes numbers not integers numbers ? .

How we can deal with this equation $a^{n}+b^n=c^{n}$ if it was gaven to have solutions in primes numbers not integers numbers ? . note: $a, b, c $ are primes . Is there someone give us a reason ...
0
votes
0answers
172 views

“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
0
votes
1answer
31 views

Existence of solution for the diophantine equation $100x - 23y = -19$

In order to solve the diophantine equation: $$100x + (-23)y = -19$$ (from here) we could use the theorem that : The diophantine equation $ax+by=c$ has solutions if and only if $gcd(a,b)|c$. ...
1
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0answers
27 views

When is the existence of rational points on an ellipse equivalent to the existence of integral points?

This question is a follow-up to my previous question. For what square-free values of $d$ is the following statement true? For all $n\geq 1$, the equation $x^2+dy^2=n$ has a rational solution if ...
2
votes
1answer
48 views

Existence of rational points on ellipses equivalent to existence of integral points?

Let $d$ and $n$ be square-free natural numbers. Is it true that $x^2+dy^2=n$ has a rational solution if and only if it has an integral solution? I know this is true for circles (i.e., when $d=1$) but ...
4
votes
1answer
147 views

When does this equation have a solutions in integers : $z^x+{(\bar{z}})^{y}=1$?

let z be a complex variable and $\bar{z}$ it conjugate such us : $z=\alpha+i\beta $ .and $x, y,\alpha,\beta$ are integers . When does this equation have a solutions in integers : ...
0
votes
2answers
44 views

Integer solutions of $m^2+(m+1)^2=n^4+(n+1)^4$

$m^2+(m+1)^2=n^4+(n+1)^4$ Integral (integer) solutions needed I could find only four, are there others -1 & 0 give both sides as 1.
6
votes
2answers
95 views

Showing that $x^3+2y^3+4z^3=2xyz$ has no integer solutions except $(0,0,0)$.

Let $x,y,z\in \mathbb{Z}$ satisfy the equation: $$ x^3+2y^3+4z^3=2xyz $$ How do I prove that the only solution is $x=y=z=0$?
4
votes
2answers
127 views

Innocent-looking Diophantine equation with smallest solution of the order $10^{50}$?

Recently someone mentioned to me that there is a diophantine equation that looks very simple and innocent, but the smallest solution involves numbers of the order $10^{50}$ or something like this. The ...
0
votes
1answer
28 views

Solution of the following system

I have following system of equations: $q = wz + h + j$, $z = f_k(h+j) + h$ All variables are non-negative integers, and $q$ and $f_k$ are known. The solution of the system is given by: $w = \lfloor ...
1
vote
1answer
32 views

Quadratic Diophantine equations on the ring of polynomials

The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient ...
8
votes
2answers
353 views

For which values of $\theta$ does this equation $x^{\cos\theta} +y^{\sin\theta }=1$ have solutions in integers?

For which values of $\theta$ does this equation $$x^{\cos\theta} +y^{\sin\theta}=1$$ have solutions in integers ? Note : $x, y$ integers, $\theta$ is real number. Thank you for your help.
1
vote
1answer
32 views

Diophantine solutions to a large geometric figure

I have a related question to one I've read today, see: Integer solutions to $2x^2+5x+y^2=19$ The integers solution are part of an ellipse, with an obvious finite number of $x$. What I would like to ...
0
votes
2answers
78 views

Integer solutions to $2x^2+5x+y^2=19$

$$2x^2+5x+y^2=19$$ Don't know how to approach the problem. Similar equations required factoring after the completing a square or a similar trick. I don't see the possibility of that here though. ...
0
votes
1answer
48 views

What exactly is a Diophantine representation?

I am interested into Diophantine equations, and I have few misunderstandings. What exactly is a Diophantine representation of some set? It is some polynomial Diophantine equation, but the thing that ...
-1
votes
3answers
77 views

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions [closed]

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions Can anyone give me the specific steps for this problem?
1
vote
2answers
96 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [closed]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
2
votes
2answers
31 views

Set of all integer solutions to a linear diophantine equation

I am trying to figure out the set of all integer solutions in terms of an appropriate number of free variables for the following: $2x_1 + 12x_2 + 3x_3 = 7$. I have found that the $gcd(2,12,3) = 1$ ...
0
votes
2answers
45 views

Find the number of possible values of $a$

Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$, and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$ Obviously, factoring, ...
5
votes
2answers
109 views

Find all solutions in N of the following Diophantine equation

$(x^2 − y^2)z − y^3 = 0$ i divide by $z^3$ and look for rational solutions of the equation $A^2 − B^2 − B^3 = 0.$ The point $(A,B) = (0, 0)$ is a singular point, that is any line through this point ...
0
votes
2answers
55 views

solve $b^2c-a^2=d^3$ with some conditions.

Solve $b^2 c-a^2=d^3$ Conditions $b^2c>a^2$,  $b$>0, $c$>0,  $a$, $b$, $c$, $d$ are rational number. Example Solution $a=108$, $b=12$, $c=849$, $d=48$ Is Solving this equation impossible?
2
votes
2answers
137 views

Solving the Diophantine equation $x^n-y^n=1001$

For all $n \in \mathbb{N}$, solve the Diophantine equation $x^n-y^n=1001$, where $x,y \in \mathbb{N}$. The cases $n=1,2$ are trivial ones. But for $n>2$ I can't find any solutions. How could I ...
1
vote
4answers
79 views

quadratic diophantine's equation in form of $y=ax^2+bx+c$

I stumbled on this on Geogebra. Actually i would like to set integers pair $x$ $y$ that fits the general quadratic form. Given $(x_1,y_1)$ and $(x_2,y_2)$ are integers pairs, i am looking for set ...
0
votes
2answers
59 views

$x^2+y^2=N$, Diophantine equation

$$ x^2+y^2=N $$ $N$ integer, Find $x,y$ integer so that the Diophantine equation is fulfilled. If $N$ is a prime number, we can calculate all solutions very fast via Gauß reduction. Is ...
0
votes
0answers
43 views

Find all pairs of positive integers $(x,y)$ : $x(x+1) = y(y+1)(y+2)$

Find all pairs of positive integers $(x,y)$ : $$x(x+1) = y(y+1)(y+2)$$ I was able to find only two pairs: $(2,1)$ and $(14,5)$ and looks like no more exists. How to prove it?
1
vote
1answer
36 views

Prove that the solutions to the system of equations are integers

Let $a, b \in \mathbb{Z}$ and consider the system of equations below: $$\begin{cases} y -2x-a =0\\ y^2-xy+x^2-b=0\end{cases} $$ Prove that $x,y\in\mathbb{Q}$ implies $x,y\in\mathbb{Z}$. I ...