Questions on finding integer/rational solutions of polynomial equations.

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1
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1answer
45 views

Find all positive integers s.t. $10^m-8^n=2m^2$

Find all pairs of positive integers $(m,n)$ such that $10^m-8^n=2m^2$
18
votes
1answer
613 views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
1
vote
3answers
46 views

Diophantine equation. Three.

Diophantine equation. $X^2+Y^2=qZ^3$ I wonder at what values ​​of the coefficient $q$ equation has a solution. And of course I wonder how she looks like a formula describing their solutions. For ...
1
vote
1answer
49 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?
1
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1answer
37 views

How to solve the system $\frac{35-12b}{a-b}= \frac bk,\frac{12a-35}{a-b}= \frac {a}{1-k}$

I am trying to solve the system of diophantine equations: $$ \begin{align*} \frac{35-12b}{a-b} &= \frac bk \\[6pt] \frac{12a-35}{a-b} &= \frac {a}{1-k} \end{align*} $$ Where $a-b\ne 0,$ and ...
0
votes
0answers
24 views

Can you write a variable as the sum of two variables?

I was seeing this question and, in the develop of an answer, a question arised: I have a variable $k\in [1,40]_{\Bbb N}$ and I want write it as the sum of one variable and something more with the ...
3
votes
2answers
266 views

Whenever Pell's equation proof is solvable, it has infinitely many solutions

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 ...
0
votes
2answers
75 views

Solving homogeneous quaternary quadratic Diophantine equation

Given the equation $w^2+x^2+y^2+z^2=wx+wy+wz+xy+xz+yz$, how does one systematically enumerate all non-negative integer solutions $\{w,x,y,z\}$?
23
votes
6answers
499 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
1
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2answers
38 views

Equation with tangent and powers

I need to solve this equation for x: $$2000 \sigma = 1 - \frac{20x}{\pi^2x^2 + 100} - \frac{2 \arctan(\frac{\pi x}{10})}{\pi} $$ $\sigma$ is a known value. I need to solve this for $ \sigma = ...
5
votes
2answers
377 views

Second degree Diophantine equations

I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know ...
2
votes
2answers
99 views

What is the density of solutions for Pythagoras' and Fermat's equation $x^2+y^2=z^2$

It is now proved that, for integer $n\geq 2$, the equation $x^n+y^n=z^n$ has integer solution only when $n=2$. When $n=2$, this equation has an infinity of solutions. My question is whether there is ...
7
votes
2answers
178 views

3 Variable Diophantine Equation

Find all integer solutions to $$x^4 + y^4 + z^3 = 5$$ I don't know how to proceed, since it has a p-adic and real solution for all $p$. I think that the only one is (2, 2, -3) and the trivial ones ...
0
votes
2answers
505 views

Using recurrences to solve $3a^2=2b^2+1$

Is it possible to solve the equation $3a^2=2b^2+1$ for positive, integral $a$ and $b$ using recurrences?I am sure it is, as Arthur Engel in his Problem Solving Strategies has stated that as a method, ...
0
votes
1answer
36 views

Return to sum of powers question.

Previously I had asked a question about a Diophantine equation linked here. I have come back to think about this question but in a different manner. So here is the set up: Let A and $a_i$ be ...
0
votes
3answers
67 views

Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$

Let $a, b, c, d \in \mathbb Z$. Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$ if and only if $a, b, c, d$ are pairwise relatively prime. I am very confused as to how I should even start this ...
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votes
12answers
240 views

System of Diophantine equations. [closed]

Quite interesting are there any ideas on solving systems of equations like these? $\left\{\begin{aligned}&a^2+b^2=c^2\\&(a+k)^2+(b+k)^2=q^2\end{aligned}\right.$ Although I recorded such ...
4
votes
5answers
2k views

Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$

How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$ And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$ In both cases, ...
1
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2answers
49 views

Efficient software implementation of $x^2+3y^2=N$

I would like to implement a solver (in C) for the Diophantine equation $x^2+3y^2=N$ for non-negative integers $\{x,y\}$ and positive integer $N$. I have read online that one has to prime factorize N ...
1
vote
1answer
58 views

When is the equation $x^2-d^n y^2 = -1$ solvable?

My goal is to prove or disprove that if $x^2-dy^2=-1$ is solvable, then $x^2-d^ny^2 = -1$ is solvable for every odd $n \geq 1$. I do know that the former is solvable if and only if the continued ...
27
votes
2answers
2k views

Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?

The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$ has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$ then this has ...
2
votes
1answer
72 views

Diophantine Equation: $x^2 + 3y^2 = 11z^2$

I am having difficulty solving the following problem: Prove rigorously that there is no integer solution for the Diophantine Equation $x^2 + 3y^2 = 11z^2$ except when $x = y=z = 0$. ...
0
votes
1answer
28 views

Prove a system of simultaneous Diophantine equations has no solution.

I've been asked to show that the system of simultaneous Diophantine equations has no solutions: $3x+6y+z=3$ $12x+3y+2z=5$ I don't even know how to approach this problem, any help would be ...
1
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2answers
38 views

Diophantine Equation with 3 Variables

Find all solutions to $2x + 3y + 4z = 5$. I know how to do it with two variables, but I'm confused on how to start this with three variables.
0
votes
1answer
26 views

Diophantine equation $(x^2-1)^2-4y^2=0$

We have the diophantine equation $(x^2-1)^2-4y^2=0$, where $x$ and $y$ are positive integers. Is the only solution $x=1$, $y=0$ or can there be infinitely many solutions?
1
vote
0answers
30 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
0
votes
1answer
32 views

Diophantus problem

I was given following problem as an example of early mathematics with the solutions. But it seems i can't understand from where they are getting the 35z^2 = 5 from in the solutions. Could someone ...
13
votes
5answers
813 views

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
0
votes
1answer
31 views

Can the fundamental solution of a Pell equation be “triangulated” given multiple known solutions?

In this question about “descent” given a single Pell solution, Will Jagy gave the [accepted] answer that, for a Pell equation $$ U^2 - dV^2 = \pm 1, \tag{$\star$} $$ there is no way to determine ...
-2
votes
0answers
32 views

how to find GENUS of a given curve [duplicate]

Could you please help me to find GENUS of $x^2 -x + y- y^5 = 0$ hyper-elliptic curve. Also, explain how to apply FALTINGS theorem to find rational roots. high regards and many many thanks...
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votes
0answers
71 views

The curve with irreducible and has genus 2

Sketch the curve $f(x,y) = x^2 -x + y- y^5$, and prove that it is irreducible and has genus $2$. And, how to conclude that, $f(x,y)$ has only finitely many rational points as well as only finitely ...
0
votes
2answers
19 views

Solve this non-linear diophantine equation?

How do you go about systematically solving a Diophantine equation of this form : $217x^2 + 496y^2 = 15872$ ? I found that $\gcd(217, 496) = 31$ and reduced that equation to $7x^2 + 16y^2 = 512$ ...
1
vote
2answers
88 views

Solve $a^2+b^n=c^2$

Let $a,b,c$ be co-prime integers >1, for all $n>2$, I need help finding the integral solutions of the diophantine equation $a^2+b^n=c^2$. I saw the result but I am curious about to how to get ...
-2
votes
1answer
53 views

Solve $ (u^n-v^n)=p(u-v)^2$ [closed]

Let $n, u, v,p $ be non-zero positive integers, if $\gcd(u,v)=1$ and $u-v>1$, find all the nontrivial solutions of the Diophantine equation:$$ (u^n-v^n)=p(u-v)^2$$
1
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1answer
39 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
0
votes
0answers
41 views

Woking Heron's Formula In Reverse

I'm writing a program to generate randomized Heron's Formula word problems. I need to figure out how to work the problem in reverse so that the answer will come out to an integer. As an example, if I ...
0
votes
1answer
34 views

Buying dogs and Cats and mice

This problem is only considering positive integer solutions: You must spend exactly 100 and purchase exactly 100 animals. Each dog costs 15 and each cat costs 1 and each mouse costs .25. How many of ...
0
votes
1answer
15 views

FLT and non-maximal orders

The ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is the unique maximal order of that field. Kummer's attempt at proving FLT fails for prime exponents which are irregular, i.e. divide ...
0
votes
2answers
28 views

Diophantine Equation Related To Triangles

a,b and c are the sides of a triangle and a, b, c are integers. I need to solve the following Diophantine equation for positive integral values of k. $bc(b+c-a) = k^{2}(a+b+c)$ I think some ...
1
vote
2answers
26 views

Find integer solutions of $(1) xy=2x+2y$ and $ (2)xy=2x+y.$

I've tried this for the first one:$xy=2(x+y).$ Therefore either x or y is divisible by 2. And I'm totally stuck on the second. How to solve these?
5
votes
2answers
232 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
4
votes
2answers
266 views

Solutions to the Mordell Equation modulo $p$

It is well known that for any nonzero integer $k$ the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions $x$ and $y$ in $\mathbf Z$, but it has solutions modulo $n$ for all $n$. One proof of ...
0
votes
2answers
45 views

Number of integer solutions of two similar equations

Find the number of integer solutions of: (a) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{20}}$$ (b) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{2014}}$$ I know the ...
16
votes
2answers
891 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as simply as possible? I have seen some ...
1
vote
2answers
57 views

How do you solve $k(a^2-b^2)=2(ax-by)$?

let $a,b,c,d,x,y,k$ be all non-zero positive integers >1. If $a^2-b^2 \neq0$,how do you find all the pairs $(x,y)$ such that $k(a^2-b^2)=2(ax-by)$. I have found so far only solutions where ...
-1
votes
1answer
43 views

Simple proof that $f(x)=x^3-3(u+v)x^2+3(u^2+v^2)x-(u^3+v^3)=0$ has no integral solutions [closed]

let $x,u,v>1$ if $\gcd(u,v)=1$, prove that $f(x)=x^3-3(u+v)x^2+3(u^2+v^2)x-(u^3+v^3)=0$ has no integral solutions.
1
vote
3answers
77 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
2
votes
0answers
46 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
1
vote
3answers
94 views

Non-linear diophantine equation

Let $k$ and $n$ be positive integers and $y(n-x)=(k+nx)$. What is the condition of $k$ and $n$ such that there exist positive integers $x, y$ as the solution of $y(n-x)=(k+nx)$?
0
votes
3answers
42 views

Show that there are only trivial solutions

How can I show that the only solutions of the diophantine equation $x^2+y^2=1$ are the trivial ones: $(x,y)=(0,1), (0,-1), (1,0), (-1,0)$ ? That's what I thought: $$x \equiv 0,1 \pmod 2 \Rightarrow ...