Questions on finding integer/rational solutions of polynomial equations.

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2
votes
1answer
106 views

Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
-4
votes
1answer
87 views

Project Euler - task №390.

When this task was not clear what the equation to be solved. This equation? $x^2y^2+z^2y^2+x^2z^2=r^2$ in integers. It is not clear, because this equation is quite simple and I do not think that ...
2
votes
2answers
52 views

Solving $a^2-24 a=k^2$ across the integers

$$a^2-24 a=k^2$$ I was trying to solve another problem that essentially boiled down to finding all possible integer values of a such that $a^2-24 a$ was a perfect square, and I thus came up with the ...
1
vote
1answer
64 views

Diophantine equation in four variables

I would like to find a parametric solution for the following diophantine equation: $-4 (1-a_1^2)(1-a_2^2) + (1+a_3^2 -a_1^2 -a_2^2)^2 = a_4^2$ Does such a solution exist? How does one go about ...
0
votes
0answers
51 views

System of quadratic diophantine equations 2

I am looking for a way to simultaneously transform the following four expressions into perfect squares, $1+x_1^2, 1+x_2^2, 1+x_3^2, x_1^2+x_2^2+x_3^2$, i.e. I want to find a rational parametrization ...
1
vote
4answers
172 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...
2
votes
1answer
99 views

Diophantine equation: $2 a^2 + 2 b^2 = c^2 + d^2$ [duplicate]

I am looking for integer solutions of the following equation: $$ 2a^{2} + 2b^{2} - c^{2} - d^{2} = 0 $$ Preferentially the solutions should obey $a+b+c+d=0$. By inspection I found the solutions: ...
1
vote
2answers
91 views

Diophantine equation with cubes.

Interested in the solution in general Diophantine equations of the form: $X^3+Y^3+Z^3=3XYZ+q$ $q$ - what some integer. Solutions similar equations can be written. Since this equation is easy, as ...
0
votes
3answers
60 views

Prove the solutions of following equation exists [closed]

$x^2+y^2=z^n$ has a solution in $\mathbb{N}$, for all $n \in \mathbb{N}$. The problem is to show that for every natural number n there exists 3 integers which show the above relation for ...
1
vote
2answers
66 views

Find all integral solutions

Find all the integral solutions of $$x^4+y^4+z^4-w^4=1995.$$ Please elaborate the solution. I tried but can't understand what to do.
3
votes
3answers
96 views

Solve the following equation in positive integers $x$ and $y$

What are the solutions in positive integers of the equation: $${1+2^x+2^{2x+1}=y^2}$$ I tried to factorize the equation but it didn't help much. Clearly $y $ is an odd integer. Substituting $y ...
1
vote
0answers
112 views

How can we prove that this equation cannot be solved?

How can we prove that this equation cannot be solved? $ 25k^3+30k^2+23k+3=x^2$ where x,k are integer numbers
-1
votes
1answer
109 views

The equation $X^{n} + Y^{n} = Z^{n}$ , where $ n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are intergers.

The equation $X^{n} + Y^{n} = Z^{n}$ , where $n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are integers. My solution: False. Because if we let $X=0 ...
1
vote
0answers
22 views

Why the proof of Catalan's conjecture is not easily generalizable?

Let $x,y>0$, $u,v>1$ be integers. Why is it easier to solve $x^u-y^v=1$ than $x^u-y^v=2$? Is there possible some group behind the first equation which has some nice property that the group made ...
1
vote
1answer
36 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
1
vote
0answers
120 views

Integral points on varieties and solutions to Diophantine equations

I am looking for a book (or article, or notes...) explaining details about the link between integral points on varieties defined as complement of certain divisors and integral solutions to the ...
2
votes
3answers
92 views

Finding integral solutions to the equation $x^4-ax^3-bx^2-cx-d=0$

How many integral solutions exist for the equation: $$x^4-ax^3-bx^2-cx-d=0\qquad a\ge b\ge c\ge d\qquad a,b,c,d\in\Bbb{N}$$ I have no idea where to begin even.Please help.
0
votes
0answers
32 views

Least squares over the integers (diophantine least squares?)

I have the following problem and I do not even know under which mathematical field I should look for an answer, so any hint is highly appreciated: Let S be the ellipsoid $$ ...
0
votes
0answers
21 views

Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
1
vote
1answer
47 views

non-negative solutions with upper boundary, in diophantine equation

I wanted to find out in how many ways I can do something, but I don't know combinatorics enough. Can you help me and show or give advice, what we should do in such situations as presented below? Let's ...
0
votes
3answers
46 views

Equation involving power of two

I want to show that the equation $2^x - 1 = 3^y$ does not have any positive integer solutions except for $ x = 2 , y = 1$ . Is it possible to prove the assertion using binary representation of powers ...
0
votes
1answer
53 views

Is there any solution to this quadratic Diophantine equation?

Can one find all positive integer triplets $(x,y,z)$ satisfying this parametric equation : $$ax^2 + (1-a)x + by^2 + (1-b)y = cz^2 + (1-c)z$$ Here, $a, b, c$ are positive integers greater than $1$. ...
0
votes
1answer
35 views

Random nonegative solution of multivariate linear Diophantine equation

Consider a diophantine equation in n variables: $a_1x_1+a_2x_2+...+a_nx_n=k$ All $a_i$'s, $x_i$'s and $k$ are restricted to non-negative integers $\mathbb{Z^+}$. (note that because of domain ...
0
votes
3answers
123 views

Curves triangular numbers.

Sometimes you have to deal with this equation: $X^2+aX+Y^2+bY=Z^2+cZ$ $a,b,c$ - integer coefficients. I wrote below - to start a particular solution of Diophantine equations. To do this, use the ...
7
votes
0answers
129 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
4
votes
3answers
223 views

Positive integer solutions of $a^3 + b^3 = c$

Is there any fast way to solve for positive integer solutions of $$a^3 + b^3 = c$$ knowing $c$? My current method is checking if $c - a^3$ is a perfect cube for a range of numbers for $a$, but this ...
0
votes
0answers
38 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
2
votes
2answers
79 views

Show that $a^2 - 15b^2 =3$ has no integer solutions.

Show that $$a^2 - 15b^2 =3$$ has no integer solutions. I'm not overly experienced with number theory nor Diophantine equations, but upon looking around a bit I've realised this is a Pell-type ...
0
votes
1answer
38 views

Solutions of Diophantine equations in general form.

Prior to that solved a similar equation. Solutions like wrote. Then I thought to solve a similar equation. Diophantine equation: $X^2+XY+Y^2=Z^2+1$ Some solutions are unpretentious ...
0
votes
3answers
31 views

Find a and b in quadratic equation

I have the problem to find $a$ and $b$ given $f(x)=-x^2-2ax+b, a\neq0$ $f(1)=3$ , and the maximum value of $f(x)$ is $4$ and have they key with the answer $a=-2,b=0$, but which steps do I take to ...
1
vote
2answers
69 views

To solve $1+2^mp^2=q^5$

How do we find all posible solutions of $1+2^mp^2=q^5$ for positive integer $m$ and primes $p,q$ ? $m=1,q=3,p=11$ is a solution , is there any other solution ?
0
votes
2answers
46 views

Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$

From research completely unrelated to Number Theory I stumbled onto the following equation: $$ xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3 $$ for $x, y, z$ integers, $x,y,z \neq 0$(I ...
33
votes
4answers
840 views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
2
votes
2answers
124 views

When is the power of a binomial equal to the sum of like powers of its terms?

Question: Under what circumstances/restrictions on $x$ and $y$ does $(x + y)^n = x^n + y^n$ given the value of $n$? That is, what can we tell about $x$ and $y$ from the value of $n$ and the equation ...
0
votes
0answers
46 views

Nature of roots of a quadratic equation

If $L+M+N=0$ and $L, M, N$ are rationals the roots of the equation $( M+N-L) x^2 +(N+L-M)x + (L+M-N) =0$ are $a)$ Real and irrational $b)$ Real and rational $c)$ Imaginary and equal ...
3
votes
1answer
51 views

Integer solutions to equations of the form $a^n+b^n+\cdots=c^n$

I shall refer to the number of terms on the left side of the equation as $m$. Suppose that all numbers in the equation are positive integers. I am wondering if anything is known about for which ...
3
votes
1answer
694 views

Fermat's Last Theorem where $n$ is a power of $2$

I have seen the proof that Fermat gives for $$x^4 +y^4 \neq z^2$$ which we know also works for $z^4$. BUT I am wondering if the same basic argument can be used for the power of $2^n$. Thinks 8,16,32 ...
1
vote
0answers
31 views

To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ on all integers

Refering To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ , I think it is an interesting question, if the possible solution are integers, thus How do we find all integers $(m,n)$ such that $ ...
1
vote
2answers
54 views

Prove/Dis-Prove that the set of diophantine equation is infinite

Given diophantine equation $4x^3 - 3 = y^2$ ($x > 0$). How many solutions are there ? I don't know where to start, please give me a hint
0
votes
2answers
34 views

A diophantine question about squares

I have been trying to solve the following problem: Classify triples of integers $(m,n,k)$ satisfying the following equation $2mn+m+n=k^{2}$. It is very easy to obtain some solutions. However, I am ...
1
vote
0answers
29 views

Another triple.

Solving the equation. $X^2+Y^2+Z^2=X^3$ got some solutions, but still the question remains. Below are all the decisions or not? $X=5t^2+2t+2$ $Y=11t^3+5t^2+2t$ $Z=2t^3+10t^2+4t+2$ And more. ...
1
vote
1answer
78 views

Solving Pell's equation: algorithm to converge $\sqrt n$

I'm trying to come up with an algorithm to solve the Diophantine equation $$ x^2 - ny^2 = 1 $$ for minimum values of $x$ when $ n $ is given. This equation is also known as Pell's Equation. The ...
0
votes
2answers
44 views

Find all solutions in positive integers of the Diophantine equation (proof explanation)

An example problem in my textbook asks: Find all solutions in positive integers of the diophantine equation $x^2 + 2y^2 = z^2$. The provided proof appears as follows: $2y^2 = z^2 - x^2 = (z - ...
1
vote
3answers
59 views

Diophantine equations problem/exercise 3

Find all the pythagorean tripples (x,y,z) with x=40. Well I started with the known formulas for the pythagorean tripples but got me nowhere. Or I was not able continue the thought process required. I ...
0
votes
1answer
63 views

Diophantine equation exercise [duplicate]

Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not ...
-1
votes
3answers
25 views

Find a and b in equation given range of x

I have the problem to find $a$ and $b$ given $-ax^2+bx+4\geqslant0$, $-1/3\leqslant x\leqslant4$ and have they key with the answer $a=3,b=11$, but which steps do I take to get to that answer?
2
votes
1answer
70 views

Find all integers n such that n−2014 and n+ 2014 are both triangular numbers.

I came across this problem when searching for triangular numbers questions. I know that I need to use the equation, $$\frac {n(n+1)}{2} $$ but I don't know how to apply it to this problem.
0
votes
3answers
130 views

Problem Heron of Alexandria.

Meaning of the problem is to find two right triangles equal perimeter, but with a predetermined magnification area. That is necessary to solve a simple system of equations. ...
2
votes
1answer
98 views

Number Theory Question: $x^2-33y^3=10$ no solutions

I've been struggling to get my head around this for a while! Show that: $x^2 - 33y^3 = 10$ has no integral solutions
0
votes
0answers
21 views

Matrices and diophantine equations

Let A be $mxn$ matrix with integral elements, and let r denote the rank of A. For $1 \leq k \leq r$, let $d_k (A)$ be the gcd of the determinants of all $kxk$ matrices. This is called determinantal ...