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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

0
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2answers
66 views

Find all values of $n$ such that $\varphi(n) = n/6$. [duplicate]

Using the product formula (the formula with the prime factors of $n$), I got $$1=6\frac{(P_1-1)}{P_1}\frac{(P_2-1)}{P_2}\cdots\frac{(P_k-1)}{P_k}\,.$$
2
votes
0answers
30 views

Finding a subspace of dimension $3$ which does not intersect a rational subspace of dimension $2$

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
3
votes
1answer
41 views

Is this Diophantine problem solvable without invoking Fermat's Last Theorem?

Let $a,b,c,n$ be positive integers with $a<b<c$ and $n\geq 3$ odd. Given that $a^n + b^n < 2c^n$, can one prove that $a^{n+2}+b^{n+2}\neq c^{n+2}$ without invoking Wiles' theorem ? Or is this ...
9
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2answers
145 views

Solutions to $a,\ b,\ c,\ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \in \mathbb{Z}$

I came across a puzzle in a Maths Calendar I own. Most of them I can do fairly easily, but this one has me stumped, and I was hoping for a hint or solution. The question is: What are the solutions to ...
0
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3answers
84 views

Pythagorean like Diophantine Equation

I am trying to solve this problem. http://www.javaist.com/rosecode/problem-527-1-2-3-type-Pythagorean-triangles-askyear-2018 Here we have to find all positive integral solution of $a^2+2b^2=3c^2$ ...
0
votes
0answers
21 views

What can I use online to plot 2D and 3D diophantine equations?

Using the algebraic operations and equality, and (scatter) plotting the integer solutions, for example: Plotting $x$ and $y$ for $x^2 + y^2 + n^2 = 9$ $(2, 2), (3, 0)$ and $(0, 3)$ are among the ...
2
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2answers
121 views

Does there exist solutions to the diophantine equation $x^3-y^3 = n$? If so, what conditions on $n$ are necessary?

Does there exist solutions to the diophantine equation $x^3-y^3 = n$? If so, what conditions must be placed upon $n$ for solutions to exist?
1
vote
3answers
87 views

Find all positive integers $a$ and $b$ such that $(1 + a)(8 + b)(a + b) = 27ab$.

Here's the problem I'm having difficulties with: Find all positive integers $a$ and $b$ such that $$(1 + a)(8 + b)(a + b) = 27ab\,.$$ Does anyone have an idea how to do this? Any detailed solution ...
2
votes
1answer
54 views

Integer equations

I have $2$ following problems. Find integer roots of $$\begin{align} &1)~\frac{x+y}{x^2-xy+y^2}=\frac3z \\ &2)~x^3y^3-4xy^3+y^2+x^2-2y-3=0 \end{align}$$ I have no idea to solve them. I try ...
3
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1answer
71 views

Solving the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$

I want to solve the Diophantine equation $y^3 = 4x^2+4x+ 5$ for $x,y \in \mathbb{Z}$. The right hand side factors as $(2x+1-2i)(2x+1+2i)$. Am I right that such a factorization can be found using the ...
0
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3answers
44 views

Showing that the Diophantine equation $3x^2 + 6y^6 + 1 = 8xy^3$ has no solutions $x,y \in \mathbb{Q}$

I want to show that the Diophantine equation $3x^2 + 6y^6 + 1 = 8xy^3$ has no solutions $x,y \in \mathbb{Q}$. I tried factoring, but didn't manage (but I'm not good at factoring). Then I tried ...
1
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2answers
54 views

Solving the Diophantine equation $y^2 = x^4+x+ 2$ for $x,y \in \mathbb{Z}$

I want to solve the Diophantine equation $y^2 = x^4+x+ 2$ for $x,y \in \mathbb{Z}$. I already found 4 solutions: $(x,y) = (1,\pm2)$ and $(x,y)=(-2,\pm4)$. It can probably be solved using some ...
0
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1answer
53 views

A simple equation with a complicated property

Let, $\Bbb{P}$ denote the set of all odd prime numbers and $\Bbb{N}$ be the set of all natural numbers. Let, $2a,2b$ be two even numbers both greater than $4$. Define, $A=\{(p,q)\in\Bbb{P}\times\Bbb{...
6
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4answers
126 views

Quick! Prove that $3^{15}+37$ is a square.

CONTEXT: I have been studying the odd powers of $3$ and trying to determine when they are "close to" square numbers; more specifically, I have conjectured that there exist finitely many solutions $m,n$...
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0answers
50 views

Trivial solutions of a diophantine equation

Let $K$ be an odd degree number field. Consider the Diophantine equation: $$ X^4 + bY^4 =Z^2 $$ where $b\neq 0$. Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
1
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0answers
38 views

Conditions on those Pell-type equations to admit solutions

I am facing those generalised Pell equations, $a^2-Db^2=-8$ and $x^2-Dy^2=8$, where $D=t^2+8t$ for an odd $t$, $t>2$. In particular, I would like to find the cases where both equations admit (...
6
votes
1answer
51 views

Find all non-negative integers $a, b$ satisfying $|4a^2 - b^{b+1}| \leq 3$

Find all non-negative integers $a, b$ satisfying $|4a^2 - b^{b+1}| \leq 3$. I have been trying a simpler case $4a^2 - b^{b+1} = 0$. I found that when $b$ is odd then $b^{b+1}$ have to be in form $b^...
2
votes
2answers
56 views

Rational solution to a system of equations

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
2
votes
2answers
62 views

Solving the homogeneous Diophantine equation $x^3 + 2y^3 = 7z^3$ for $x,y,z \in \mathbb{Q}$

I want to solve the homogeneous Diophantine equation $x^3 + 2y^3 = 7z^3$ for $x,y,z \in \mathbb{Q}$. First note that $(x,y,z) = (0,0,0)$ is a solution. For further solutions it suffices to search ...
6
votes
1answer
75 views

Solving the Diophantine equation $y^2 = 4x^3 + 1$ for $x,y \in \mathbb{Z}$

I want to solve the Diophantine equation $y^2 = 4x^3 + 1$ for $x,y \in \mathbb{Z}$. Note that $y$ is odd, since $y$ even would give a contradiction $\mod{2}$. Hence $\frac{y-1}{2}, \frac{y+1}{2} \in \...
0
votes
0answers
29 views

given $p,q,r \ge 3$ study the diophantine equation $x^py^q=z^r-1$ using the $abc$-conjecture

I want to show that given $p,q,r \ge 3$ the diophantine equation $x^py^q=z^r-1$ has only finitely many solutions with $x,y,z \in \mathbb{N} = 1 ,2, \dots$ assuming the $abc$-conjecture. The proof ...
2
votes
4answers
73 views

find all $k$ such that $k^5+3$ is divisible by $k^2+1$

That's it. I have a solution with substitutions, but it's tedious and not too generic. Is there a solution using some cool theorem for polynomials divisibility?
1
vote
2answers
37 views

Use of diophantine equation (money problems, 3 variables)

My friend has in his wallet some notes of $\displaystyle{ 20 }$ , $\displaystyle{ 50 }$ and $\displaystyle{ 100 }$ euros. He has $\displaystyle{ 15 }$ notes and the total value of them is $\...
4
votes
1answer
325 views

Does this polynomial have a rational value which is the square of a rational number?

I have the following polynomial: $$P(x,y,z):=9y^2z^2-30x^2z+90xyz+54yz-270x+81\in\mathbb Q[x].$$ It came up in a larger proof, and I would need in order to complete the proof to prove the following ...
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0answers
63 views

Derivation of the Frey curve for $a^2+b^6=c^p$

At the bottom of page $5$ in Bennet Chen's paper, they give the Frey curve for $a^2+b^6=c^p$ as: $Y^2=X^3−3(5b^3+ 4ai)bX+ 2(11b^6+ 14ib^3a−2a^2)$ Can anyone tell me how they derived this Frey curve?
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votes
4answers
42 views

Diophantine equation: solving $a^2+4n=b^2$

I found myself working with diophantine equations but I have no experience at all with them. Given an integer $n$, can I find two integers, $a$ and $b$, such that $$a^2+4n=b^2$$ How would you guys ...
3
votes
1answer
75 views

Solve equation $n^4+n^2+1=p$, where p is prime number

If $n \in \mathbb N$ and $p$ is prime number solve equation $n^4+n^2+1=p$ i can write that like this $$n^4+2n^2-n^2+1=p$$ $$(n^2+1)^2-n^2=(n^2+1-n)(n^2+1+n)=p=1 \cdot p$$ Since $n^2+1+n>1$ then $n^...
5
votes
5answers
92 views

If $2n+1$ and $3n+1$ are perfect squares, then prove that $8|n$.

If for some number $n\in \mathbb N$, the numbers $2n+1$ and $3n+1$ are perfect squares of integers, then prove that $8|n$. if $2n+1=m^2$ and $3n+1=k^2$ then $k^2-m^2=3n-2n+1-1=n$ now I need to show ...
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0answers
18 views

equation for integers

ok here is the problem. suppose i want to find all pair of a and b such that a*b=10, let's this was first equation in our system and now i need some second equation, such that after putting it into ...
0
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0answers
30 views

Chinese remainder theorem and Diophantine equation implementation

I needed an advise on implementing and solving one problem and the others like. I came across two sentences that I think will be helpful. Those are the Chinese remainder theorem and the Diofantic ...
1
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1answer
72 views

Find the number of non-negative integer solutions to linear systems

For instance with two variables: $ax + by = c$, where x and y are variables. I found these two threads [1, 2], where the solution is equal to $\binom{n+p-1}{p-1}$, where n is the desired sum and p is ...
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0answers
88 views

Fermat-Catalan eighth powers

There are two Fermat-Catalan solutions that have as an eighth power in their addend the numbers, $33^8$ and $44^8$. In Darmon and Granville's paper, they show that the generalized Fermat Equation has ...
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0answers
72 views

$0<|\sqrt a-\sqrt[3]b|<\epsilon$ for $a,b\in\Bbb Z_+$

I'm trying to solve the following problem: Given $\epsilon>0$, are there positive integers $a,b$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$ ? My solution: given $n\in\Bbb N$, $$|\sqrt{n^2}...
1
vote
1answer
17 views

Proving a linear Diophantine system has no solutions

I've been tacking a problem in my abstract algebra assignment which wants me to show that the ideal $(4 + \sqrt{-5}, 1 + 2\sqrt{-5})$ is not principle in $\mathbb{Z}[\sqrt{-5}]$ and I've reduced it to ...
1
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1answer
49 views

Number of solutions in linear equation with 3 variables

Is there a way how to determine number of solutions in linear equation like this: $ax + by + cz = d$, where $a,b,c,x,y,z,d$ are non-negative integers and $a,b,c,d$ are known?
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3answers
70 views

Solutions for $2^n+1=p^q$

I have a problem I can’t solve, please help! Find all positive integer triples $(n,p,q)$ satisfying $2^n+1=p^q$, where $p,q>1$. There is a similar problem I can solve: Prove that it is not ...
9
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0answers
216 views

$m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. [closed]

Find integer numbers $m$, $n$ such that $m^{5}n+7$ and $n^{5}m+7$ are cubes of integer. I guess the only pair is $(1;1)$, but I don't know what way we should follow to solve this problem. Help me ...
6
votes
2answers
194 views

General way of solve $ax^2+by+c=0$

For example ,the diophantine equation $$x^2+1=25y$$ we can solve this by finding particular solution $(x,y)=(7,2)$ and using this , we can get general solution. My question is "To solve $ax^2+by+c=0$...
0
votes
0answers
29 views

Decidability of $\forall\exists$ diophantine equations

By saying $\forall\exists$ diophantine equations I mean sentences of the form: $\forall x\exists y\,[p(x,y) = 0]$ where $p$ is a polynomial on $x,y$, and both $x,y$ range over natural numbers. I want ...
0
votes
1answer
73 views

Prime solutions to a congruence modulo a semi-prime

Let $p$ and $q$ be primes. Besides $\{3,13\}$ and $\{13,61\}$, find other solutions $\{p,q\}$ to the congruence $$ 1+ p+q+p^2+q^2 \equiv 0 \pmod {pq}$$ or show that there are none.
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1answer
64 views

Proving $3x^{10} - y^{10} = 1991$ has no integral solutions. Check my proof.

The problem is Prove that $$3x^{10} - y^{10} = 1991$$ has no integral solutions. I have written this proof, of which I am not sure. Assume that $x$ and $y$ are integers. $$\begin{align} 3 x^{10}...
2
votes
2answers
71 views

Solve equation in prime numbers

Solve the equation in prime number $$p^3+q^3+1=p^2 q^2.$$ I have found the solutions $(2,3), (3,2)$ and need to prove that there are no other solutions. I think that there is an inequality $p^3+q^3+1\...
0
votes
1answer
66 views

Prove that $x² + y² = z^n$ has a solution in $\mathbb{N}$, for all $n$ belonging to the set of natural numbers [duplicate]

Prove that $x^2 + y^2 = z^n$ has a solution in $\mathbb{N}$, for all $n$ belonging to the set of natural numbers. I think it uses Induction as a method of proof but just can't proceed with three ...
1
vote
0answers
23 views

Non-Linear Diophantine Equation in Two Variables [duplicate]

How many solutions are there in $\mathbb{N}\times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}$ ? I could solve till I got to the point where $1995^2$ is equal to the ...
0
votes
3answers
47 views

Hey, I got the Diophantine equation $3x + 4y = 5$ over $\mathbb{Z}$ numbers [closed]

I've got a Diophantine equation: $$3x + 4y = 5$$ over $\mathbb{Z}$ numbers. The solution for this equation on WolframAlpha is: $$x = -4n-1, y=3n+2$$ and $n \in \mathbb{Z}$. I wonder where this ...
0
votes
0answers
22 views

More information on Diophantine equations of type $axy + bx + cy = d$ [duplicate]

This question is similar to the one on the site, but it IS NOT the same. I need bibliographical references, to read more on the subject. I'm trying to find the formula for the general solutions for ...
3
votes
2answers
41 views

Basic questions about pythagorean triples and “n-lets”

I've had some difficulties finding answer to the two following questions: 1) Given one of natural numbers $a,b$ where $b$ is even and $a^2+b^2=c^2$ is there only one such a pythagorean triple? 2)How ...
1
vote
3answers
50 views

Number of solutions in degree four

Find number of postive integra solutions of the equation $ x^4+ 4y^4 + 16z^4 +64= 32xyz$. I could just proceed till that x cant be odd.
2
votes
0answers
63 views

Calculate the number of nonnegative integer solutions of $ax+by\leq c$.

If $a$, $b$, and $c$ are known, and $x$ and $y$ are integers greater than or equal to zero, how many possible values of ($x$, $y$) exist that satisfy the equation $$ax + by \le c\,?$$ I have ...
0
votes
1answer
53 views

When given a non-multiple of $3$, $k$, is it possible construct $m<k$ with these conditions? [closed]

This is another Collatz-related problem about trying to represent a number in a certain form. As is usually the case with the Collatz conjecture, this is probably not useful. My question is : Can ...