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Questions tagged [diophantine-equations]

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

2
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2answers
67 views

Variation of Pell's equation

Is it possible to find all integer solutions of the Diophantine equation $$ 2x^{2} - 3y^{2} = 5 $$ I think we have to use $\mathbb{Q}(\sqrt{6})$ somewhere, but I don't know how to use units of $\...
2
votes
1answer
40 views

A Generalised Diophantine Conjecture

I submitted without proof (and verification), the following conjecture: The Diophantine equation $$\sum_{k=1}^{L} (x_k)^n = y^n$$ has integer solutions only for $n\le L$. Fermat’s Last Theorem is ...
9
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0answers
159 views

An Engineer sets out to Prove Fermat's Last Theorem …

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. ...
3
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2answers
57 views

Uniqueness of pair $\left(a,b\right)$ in writing positive integer $V$ as $V=a^2+ab+b^2$ with $a, b \in \mathbb{N}$

Given a positive integer $V$ that can be written as $V=a^2+ab+b^2$ with $a, b \in \mathbb{N}$ and $a \geq b$, is it possible to show that the pair $(a, b)$ is unique (ie that there are no other pairs $...
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2answers
29 views

Deduction for joining two non linear equations

I have, as dummy in math, to make the deduction for joining two nonlinear equations. I have one equation $$Y_v = 0.0000464 + x_1^{1.61} + x_2^{1.24}$$ that expresses $Y_v$ as a function of $x_1$ and $...
3
votes
0answers
41 views

Find all the solutions $(x,y,z)$ $\in \Bbb Z^+$ such that $(x+1)^y-x^z=1$ [duplicate]

Find all the solutions $(x,y,z)$ $\in \Bbb Z^+$ such that $(x+1)^y-x^z=1$. This is an old problem from a math olympiad in Venezuela, in the year 2000. I don't know how to start solving this kind of ...
4
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2answers
87 views

Diophantine Equation Equaling a Square

I was wondering if someone could explain the conditions for which the general Diophantine equation $$m^2 = n^k + n^{k-1} + ... + n^1 + 1$$ where $m^2 \in \mathbb{Z}^+$ is a perfect square. For ...
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1answer
50 views

Solve the following Diophantine equation [duplicate]

I need help with solving the following diophantine equation: $$x^2+y^2=2018$$ Thanks a lot in advance!
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vote
2answers
33 views

Given surface area of a cuboid, find it's integer side lengths.

Its a rather small problem once you've boiled it down to: $2(ab + bc + ca) = 100$ (the surface area of said cuboid) Now, I'm left with the equation: $ab + bc + ca = 50$, $3$ variables and $1$ ...
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1answer
53 views

$x^3=2y^3+4z^3$ in the set of integers [on hold]

If $x, y, z$ are integers solve: $$x^3=2y^3+4z^3$$
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0answers
60 views

Fastest way to check whether $ax+by+cz=d$ has positive integer solutions, exact solutions are not required

What is the fastest way to check whether $ax+by+cz=d$ has positive integer solutions, exact solutions are not required. We know if $\gcd(a,b,c)$ is not a factor of $d$, it does not have solutions, ...
0
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1answer
53 views

General Quadratic Diophantine Equations of Three Variables

For a quadratic diophantine equation of two variables, $Ax^2+Bxy+Cy^2 =D$, it's not difficult to find the solutions as it is a generalized Pell equation. However, what happens when we incorporate more ...
3
votes
3answers
59 views

Solution to the Diophantine Equation $n^2+n=2(m^2+m)$

The other day, a friend asked if it is possible to halve a triangular number and be left with another triangular number (in fact, she asked a more geometric question, about cutting an equilateral ...
4
votes
1answer
75 views

Inverse Fermat's theorem

Wiles proved that Fermat's last theorem is true, but... does it stand for inverse case? Does equation $\frac{1}{x^n}+\frac{1}{y^n}=\frac{1}{z^n}$ have no whole number solutions for $n>2$?
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5answers
140 views

Solving simultaneous (non linear) integer equations (a bit like conics)

I'm looking for all solutions, (x,y,s,t) in the integers, for the two simultaneous equations... $$ 7x^2 - y^2 = 3s^2\\ 7y^2 - x^2 = 3t^2 $$ I have two solutions $(x,y,s,t) = (2,1,3,1)$ and $(751,422,...
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votes
3answers
76 views

Given prime $p$, find solutions to $x^2 + p y^2 = z^3$

For a prime $p$ consider non-zero integers $x,y,z$ that satisfy: $$ x^2 + p y^2 = z^3$$ Does this fit in a known class of Diophantine equations that have been studied already? I'm not sure how to go ...
1
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1answer
65 views

A prime number of the form $10^n+1$ [closed]

Find all positive integers $n$ and prime $p$ such that $$10^n+1=p$$
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0answers
19 views

Counting solutions of a diophantine equation

I am interested in the number of solutions of $$x^k - y^k = m,$$ for $x$ and $y$ less than a bound $M^{1/k}$. For $x \neq y$, this is claimed to be bounded by a multiple of $m^\varepsilon$. I do ...
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1answer
114 views

Fermat's last theorem short proof attempt

Fermat's last theorem states: (1) $x^n + y^n = z^n$ has no solutions for x, y, z and n positive coprime integers and n > 2. An open question is whether there exists a simple proof hinted at by ...
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3answers
60 views

How to get only positive solution of a system of 4 variables and equations?

I have a system of $4$ equations in $4$ variables: \begin{align} x_1 + y_1 &= m\\ x_2 - y_1 &= n\\ x_1 - y_2& = o\\ x_2 + y_2 &= p\end{align} $x_1, y_1, x_2, y_2$ are ...
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1answer
50 views

Explicit expression for solutions $(x,y)$ of Diophantine equation $ax+by=d$.

Given $a,b\in\mathbb{Z}$. It is known that $\gcd (a,b)=d$ implies $$\exists x,y\in\mathbb{Z}, \ ax + by=d .$$ I have been looking for an explicit expression of any solution $(x,y)$, in terms of $a,b,...
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2answers
51 views

Integer solutions to a Diophantine Equation

How would I find all integer solutions algebraically for $$x^2-2xy+n=0$$ if $n$ is known and a solution is known, but without factoring $n$. Also, is it possible to know how many integer solutions ...
0
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2answers
71 views

Solve $73x-137y=0$, $x,y\in\mathbb{Z}.$ [duplicate]

I want to show that the diophantine equation does only have the trivial solution $x=y=0$. Since $\text{gcd}(73,-137)=1|0$ this is solveable. So \begin{align} 137&= 1\cdot73+64\\ 73 &= 1\...
9
votes
0answers
330 views

$5 \times 5\;$ “square additive set”

Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets $A = \{a_1,a_2,\ldots,a_n\}$, $B = \{b_1,b_2,\ldots, b_m\}$ such ...
1
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0answers
130 views

How does this elementary method of solving Pell's equation compare with the classical methods?

The equation $x^2-dy^2=1$ will be transformed into a quadratic equation with the use of triangular numbers. The motivation to use the triangular numbers to express the squares $x^2$ and $y^2$ comes ...
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votes
1answer
47 views

Integer solutions of $3^x = 2 + y^2$ [closed]

$$3^x = 2+y^2$$ Solve for the integer values of $x$ and $y$. Can this be solved using graph theory and calculus? I am trying out for half an hour and uses most of my preknowledge but still it is not ...
0
votes
0answers
50 views

Rational solutions of $k^4(x^2+4)-4xk^2=z^2$, where $k$ is given rational.

How to generate some parametric family of rational solutions to $k^4(x^2+4)-4xk^2=z^2$, where $k$ is given rational.
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0answers
22 views

Finding rational solutions of a system of quadratic diophantine equations

I have a system of quadratic diophantine equations $$ x^T A_i x = b_i \text{ for all } i \in \{1, \ldots, n\} $$ where $x \in \mathbb{R}^d$ is the free variable and $A_i \in \mathbb{Z}^{d \times d}$ (...
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1answer
47 views

Integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer.

Can we generate some parametric family of integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer. I don't even know if there are finite or infinite number of solutions. For $k=7$, ...
0
votes
2answers
52 views

Rational solutions of $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.

How to generate some parametric family of rational solutions to $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.
2
votes
3answers
120 views

$a + b = c + c$

So I have the following problem: $a^2 + b^2 = c^5 + c$. I want to prove that the equation has infinitely many relatively prime integer solutions. What I did first was factor the right side to get: $$...
1
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1answer
45 views

Methods for solving Elliptic curve over Q taking advantage of Complex Multiplication

In "An Introduction to the Theory of Numbers" by Hardy and Wright, they tantalizingly introduce a bunch of properties of elliptic curves, including the possibility of having Complex Multiplication, ...
2
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2answers
98 views

$p$ is a prime iff there exist a unique $m, n\in \mathbb{N}$ such that $\frac{1}{p}=\frac{1}{m}-\frac{1}{n}$

I got this question in one of the whatsapp group I am in: $p$ is a prime iff there exist a unique $m, n\in \mathbb{N}$ such that $$\frac{1}{p}=\frac{1}{m}-\frac{1}{n}$$ Does this question even ...
3
votes
2answers
65 views

Prove there are infinitely many (x, y, z) positive integers satisfying $x^5 + y^7 = z^9$

Prove there are infinitely many (x, y, z) positive integers satisfying $x^5 + y^7 = z^9$ I have reduced the problem to finding only one solution $(x_0,y_0,z_0)$ and then using the fact that there are ...
3
votes
1answer
68 views

Requirements for an integer root of cubic equation

If we have the quadratic equation $$a x^2 + b x + c = 0$$ with $a,b,c$ integers, then a requirement for $x$ to have an integer solution is for $b^2 - 4ac$ to be a square integer. This condition is ...
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vote
3answers
91 views

What are these problems called, what progress has been made on them? $x^a + y^b = z^c$ and $ax^n + by^n = cz^n$

I was thinking about some generalizations of Fermat's Last Theorem, and I'm sure they have been studied before. The first one is looking for rational solutions of $$x^a + y^b = z^c$$ and trying to ...
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0answers
48 views

System of Linear Equations with the solution(s) being permutations of a given set

Consider the system of equations $$x_1 + x_2 + x_5 + x_6 = 26 \\x_2 + x_3 + x_7 + x_8 = 26 \\x_3 + x_1 + x_9 + x_4 = 26 \\x_4 + x_5 + x_{10} + x_{12} = 26 \\x_6 + x_7 + x_{10} + x_{11} = 26 \\x_8 + ...
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0answers
29 views

Find the Farey triple given the base vertex of any isosceles triangle in Farey diagram.

There is a visualization of the circular Farey diagram where all triangles are isosceles. Observe that any rational $\frac{a}{b}$ distinct from $\frac{0}{1}$ and $\frac{1}{0}$ is always the vertex ...
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4answers
68 views

Rational solutions of $2(1+x^2)+4x(y-y^3)^2=z^2$

Does there exist a non-trivial rational solution of $2(1+x^2)+4x(y-y^3)^2=z^2$. This equation might seem very uninteresting to many of you but it has resulted after solving many simultaneous ...
3
votes
0answers
58 views

Sum of numbers on dice using stars and bars

I derived a general result about the sum of numbers on a die whose correctness I was wondering about, so could someone please confirm it? It is as follows: If $k$ dice are thrown, and the sum we ...
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0answers
13 views

Upper bound for the sum of two Frobenius Number from two specified sequences of numbers

I have a sequence of primitive numbers $A=\{a_1,...,a_m\}=\{a_1,...,a_{m-1},n\}$, i.e: $gcd(a_1,...,a_m)=1$, where $a_1 < a_2 < ... < a_m$. Also, I have the second sequence of numbers: $B=\{...
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votes
4answers
141 views

Showing that $x^2+5=y^3$ has no integer solutions.

I'm trying to show that the Diophantine equation $x^2+5=y^3$ has no integer solutions using the fact that $\mathbb Z[ \sqrt{-5}]$ has class number two. I think I have the general idea, but I'm having ...
2
votes
1answer
34 views

Under which conditions is $\gcd(a+bx,c)=1$ solvable and what is the solution?

Let $a,b,c\in\mathbb{Z}$, $c\neq0$. When is $\gcd(a+bx,c)=1$ solvable and what is $\{x\in\mathbb{Z}\mid\gcd(a+bx,c)=1\}$? A sufficient condition appears to be $\gcd(a,b)=1$ but it is not necessary as $...
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3answers
86 views

Distinct integer solutions to the equation $(x^2+y^2+z^2)/(x+y+z)=2n$ [closed]

I’ve been set this problem by my teacher, and had no idea where to start. $x, y, z$ and $n$ must be unique positive integers. I’ve found solutions where there is an equality for any of $x, y$ and $z$ ...
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0answers
50 views

Level Lowering obstructions

For the generalized Fermat equation: $a^p + b^q = c^r$ with $p,q,r\ge3$ and $\gcd(a,b,c)=1$ one can construct the Frey Curve: $y^2=x(x-a^p)(x+b^q)$ which is semi-stable for those primes $m$ ...
2
votes
1answer
83 views

solution of an algebraic equation with integers [closed]

Prove that the equation $x^2+y^2+z^2=(x-y)(y-z)(z-x)$ has an infinite number of solutions when $x,y,z$ are integers. I started from specific cases.
9
votes
3answers
1k views

Let k be an integer. Disprove: “The equation $x^2 − x − k = 0$ has no integer solution if and only if $k$ is odd.”

My problem is I keep ending up proving the statement true, instead of disproving it. I was getting it mixed up in my mind so I broke it down into very explicit steps but now I'm wondering if I'm ...
2
votes
1answer
42 views

Resolving a rational system of equations with too many unkowns

A little bit of context. While working a larger proof (the proof is quite related to this question I asked), I stumbled upon the following problem. The question. Can we find $x_1,\ldots,x_{24}\in\...
-4
votes
1answer
50 views

Fifth root of $x$ raised to three… [closed]

Find the values for $x$ and $y$ if the following equations are true. \begin{align} \sqrt[5]x^3 &= y^2 - 2\\ x - y &= 7y \end{align} The values of $x$ and $y$ are positive integers not more ...
1
vote
0answers
16 views

LTE applications

Let k be a positive integer. Find all positive integers n such that $3^k | 2^n - 1$ I just started reading and learning about the Lifting the Exponents Lemma, and I want to try and use it in this ...