Questions on finding integer/rational solutions of polynomial equations.

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5
votes
2answers
92 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
26 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
64 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
41 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
74 views

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

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
79 views

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

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!!!
-4
votes
1answer
25 views

How to use TI-Nspire CX CAS to solve Diophantine equation? [on hold]

Such as 58x+75y=1, please tell me the command(of ti-nspire). I don't know how to define a variable as an integer. I want the general formula of the solutions.
2
votes
2answers
23 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
40 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
86 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
52 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?
3
votes
2answers
124 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 ...
0
votes
4answers
70 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
56 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
38 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
32 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 ...
3
votes
2answers
56 views

Can $a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$ be an integer if $x$, $y$, and $z$ are not squares?

Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$ $$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$ ...
7
votes
1answer
101 views

The Diophantine Equation $x^2+y^4=2z^4$

We know that the Diophantine equation $x^2+y^4=2z^4$ has infinitely many solutions . Some of them are shown below $$(y,z)=(1,1),(1,13),(1343,1525),(2372159,2165017).$$ I investigated the ratio of ...
4
votes
1answer
79 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...
5
votes
2answers
67 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
4
votes
4answers
567 views

A 3rd grade math problem: fill in blanks with numbers to obtain a valid equation

Even though this is a 3rd-grade math problem, people found it extremely hard. Any people have a solution, or algorithm is welcome. I'll try make a program base on the algorithm and see if it's ...
-2
votes
1answer
40 views

Solving the Diophantine equation $x^2-y^2=a$, $\{x,y,a \in \mathbb Z^+\}$ [on hold]

the diophantine equation is a simple case divider equation or Pell's equation?
0
votes
1answer
30 views

Diophantine equations in $\Bbb Z$ [duplicate]

$x + 2y + 3z = 4$ $w = x + 2\times y$, then the equation becomes $w + 3z = 4$. $\gcd(1, 3) = 1 | 4$, so this two variable equation is solvable. $w = -2, y = 2$ i can't seem to pass this point
3
votes
1answer
71 views

Solving a Diophantine equation3

The Diophantine equation that I have to solve is: $$343x^2-27y^2=1$$ This question has already been posted by other user but it has not received an answer. I proved to solve it. This is my attempt: ...
1
vote
3answers
53 views

Is the following inequality true $(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$?

Let $a,b,c$ be all positive integers greater than $1$. If $$a>b^2$$ and $$a^3-b^6> 3c$$ Is this the following inequality true?: $$(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$$ I tried to ...
0
votes
0answers
29 views

Find all solutions to the Diophantine equation or show that none exist [duplicate]

The equation is $17x^4 + 5y^4 = 35z^4$ I reduced $\pmod 5$ but that just told me $x$ has to be a multiple of $5$. Not sure where to go from here. Any help would be appreciated. I've only taken an ...
7
votes
2answers
161 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
0
votes
0answers
75 views

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ?

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ? Please help . Thanks in advance
1
vote
2answers
56 views

Does the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ?

How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers ...
3
votes
1answer
47 views

Cubic Congruence Solutions

While I was reading a paper on number theory, there was a claim which wasn't prove there and I couldn't find a way to justify it. The claim is as follows For a prime $p$, when $p\nmid a$, the number ...
6
votes
4answers
765 views

Correct statement of Fermat's Last Theorem

I'm looking at the wikipedia page on Fermat's Last Theorem In the statement it requires $a,b,c$ to be positive integers. Is that correct? I always took it to be no solutions in non-zero integers. ...
1
vote
1answer
72 views

Solving to find the general equation with a “mod” equation

They probably aren't called "mod" equation but i couldn't think how else to word them, so I have this equation $8x + 10y ≡ 8 \pmod 7$ And have been tasked with finding the general solution, I know ...
0
votes
1answer
21 views

Integer solutions of a degree 3 curve

Suppose you have a square pyramid made out of rigid balls and all these balls are equal. Suppose now that you want to fill a square with the same number of balls that the pyramid is made. If $x$ ...
1
vote
0answers
24 views

Would my paper be considered Diophantine?

From Wikipedia In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is ...
2
votes
2answers
47 views

Types of elliptic curves

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 ...
3
votes
1answer
38 views

Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
0
votes
0answers
54 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
-6
votes
3answers
65 views

Solve the equation $2xy+2x-5y=40$, if $x$and $y$ are whole numbers. [closed]

Solve the equation $2xy+2x-5y=40$, if $x$ and $y$ are whole numbers. Could anyone give me a step by step answer?
0
votes
1answer
74 views

equation $x^4 + y^4 = z^4$

Diophantine equations that are insoluble in $\mathbb{Z}$ may become soluble in finite integral domains. Show that \begin{equation*} x^4 + y^4 = z^4 \end{equation*} is soluble (as a congruence) in ...
1
vote
2answers
64 views

How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
4
votes
1answer
80 views

30th problem of the fifth book of Diophantus;

Is there a complete answer to this problem? I have found Saunderson's answer, but I believe it is missing a few answers. The problem states: $a^2+b^2=d^2 \\ a^2+c^2=e^2 \\ b^2+c^2=f^2$ Saunderson ...
2
votes
0answers
41 views

Diophantine eqution with odd prime

HOW to find all possible set of solutions of an equation type $y^p \pm 2 = x^2$, where $p$ is any odd prime High regards to one and all
1
vote
0answers
60 views

Determine all natural numbers n and m that satisfying in this equation. [closed]

I'm trying to solve the following question: Determine all natural numbers n and m such that: $$n ^ { n ^ n } = m^m.$$ I don't have any idea about this question. Can somebody help me or give ...
2
votes
1answer
56 views

Solvability of the equation $2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

As a natural extension of the question titled Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?, I wonder if the equation $$2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + ...
11
votes
0answers
75 views

Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

It is known that given primitive (co-prime) integer solutions to, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$ then there is one $x_i$ such that $z^4-x_i^4$ is divisible by $d_4=5^4$. Additionally, Ward ...
0
votes
1answer
50 views

Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.

I successfully thought of bounding our expression examining consecutive squares that attain values close to it, and this led to the solution I'll post as an answer, which was the one reported. ...
0
votes
2answers
46 views

Solving the equation in natural numbers

How can I find the solutions in natural numbers for the following equation? $$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b$$ Where $x_{1},...,x_{n}$ are unknown. I want to find the whole of solutions ...
1
vote
2answers
60 views

Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

Aware of a Darmon-Merel theorem that asserts that if $n \geq 5$ is prime then the equation $a_{1}^{2} = a_{2}^{n} + a_{3}^{n}$ has no solution in relatively prime integers $a_{1}, a_{2}, a_{3},$ I ...
0
votes
0answers
45 views

prove that the number of solutions is finite

Prove that \begin{equation*} \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots +\frac{1}{x_n}=1,~∀i,x_i\in \mathbb{Z^+} \end{equation*} has a finite number of integer solutions. I tried to solve ...
4
votes
2answers
79 views

Find a non-trivial solution for the Diophantine Equation $17a^4 + 5b^4 = 35c^4$, or show that no non-trivial solutions exist

This is a problem on my practice exam for number theory, and we haven't had an example like this in class yet. The question is looking for a solution in $\mathbb{Z}$ for $a,b,c \in \mathbb{Z}$. I've ...