Questions tagged [diophantine-equations]

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

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Solve a Diophantine Equation $x^2+3y^2=4z^3$ [closed]

Let $x^2+3y^2=4z^3$ such that $x,y,z$ are all positive integers. I observed a case that,when x is even or odd and y is odd or even is not possible. I find some solutions by Trial & Error $(x,y,z)=(...
Guruprasad's user avatar
7 votes
2 answers
318 views

$x^5=y^2+10$ has no solutions

I am looking for an elementary way to show the equation $x^5=y^2+10$ has no integer solutions. I have checked the equation mod $n$ for $n<1000$ and it had solutions every time. Here is my proof, ...
ForgeBloyb's user avatar
0 votes
0 answers
82 views

Find a solution of a Diophantine equation $x^3+y^3=13z^3$ [duplicate]

Let $x^3+y^3=13z^3$ such that $x,y,z$ are all non zero integers. I find out one solution by Trial & Error: $(x,y,z)=(2,7,3)$. Are there any other solutions? Is there any parameterization that ...
Guruprasad's user avatar
2 votes
0 answers
61 views

If $\sqrt[n]{x + \sqrt{y}} + \sqrt[n]{x - \sqrt{y}}$ is an integer, can we always denest $\sqrt[n]{x + \sqrt{y}}$ as $(p + \sqrt{q})/2$?

I'll use $\sqrt[k]{\cdot}$ to denote the principal real $k$-th root of a real-valued input, i.e. the maximum real $k$-th root if it exists and undefined otherwise. Consider integers $n, x, y, z > 0$...
crb233's user avatar
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3 votes
1 answer
107 views

Integer Solutions to the Equation $(n-1)(x+1)(y+1)(z+1)=nxyz-1$

How can I find all integer solutions for the equation $$(n-1)(x+1)(y+1)(z+1)=nxyz-1$$ for any given positive $n$ where $n≤x≤3n-2$ and $x≤y≤z$? All attempts by me to solve this problem have so far come ...
aviolette's user avatar
-1 votes
0 answers
36 views

Existence of solutions in diophantine equations

Recently I've wondered if one equation in integers had a solution and faced a stunning (as for me) question. There is a method of proving that equation doesn't have solutions by looking at it over the ...
cooki's user avatar
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7 votes
2 answers
351 views

How to prove that the Diophantine equation $(x+y)(x+y+2)=10xy$ has no positive integer solutions

How to prove that the Diophantine equation $$(x+y)(x+y+2)=10xy\quad (1)$$ has no positive integer solutions First attempt as below is wrong :is to rewrite this equation as $$y^2 + y(2-8x) + x^2 + 2x ...
Noname's user avatar
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-1 votes
0 answers
35 views

How can I prove this statement about a Diophantine relation?

I am after a proof of the following Diophantine relation. Below I have the proof constructed so far. This is not for an assignment, just recreational math and adult learning. I would like to know if I'...
CommaToast's user avatar
10 votes
3 answers
352 views

Integer Right Triangle with Repdigit Area

This question was inspired by a tweet by Cliff Pickover. I'm seeking a right triangle with integer sides, and whose area is a repdigit number greater than $666666$. I know that Pythagorean Triples can ...
DreiCleaner's user avatar
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1 vote
0 answers
49 views

Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
tomos's user avatar
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3 votes
1 answer
109 views

Diophantine equation involving prime number

Given prime number $p$ and $n \in\mathbb{N}^*$ such that $n>p$, such that:$$(3\sqrt{p+n}+3p^2-n)(\frac{3n+1}{p}+\frac{1}{n})=3p(3n+1)+8(\frac{n}{p}+1)$$ identify the numbers $n$ and $p$. Here's how ...
fikooo's user avatar
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1 vote
0 answers
62 views

Proof of a "Fermat Last Theorem" subcase

Let us consider the equation $x^p + y^p -z^p = 0$, where $p$ is some prime number and $x,y,z$ are positive, pairwise coprime, integers. By Fermat's Little Theorem, we have that $$x^p \equiv x \space (\...
Juan Moreno's user avatar
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0 votes
1 answer
50 views

Diophantine equations - is my reasoning valid here?

I was thinking about the Diophantine equation $20x+24y=2024$ for a class today. There is an obvious solution: $x=100,y=1$ but there are other solutions as well where $x$ and $y$ are both positive ...
Red Five's user avatar
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7 votes
3 answers
340 views

Diophantine equation $2x^4 + 2x^2 y^2 + y^4 = z^2$

I've come upon the following question, and I'm a bit stumped. Any help would be greatly appreciated. Find all integer solutions $(x,y,z)$ to the equation $$ 2x^4 + 2x^2 y^2 + y^4 = z^2 $$ Here is the ...
user387394's user avatar
0 votes
1 answer
104 views

How to find multiple solutions for 3 variable, 2 degree Diophantine equation? [duplicate]

I have the equation $x^2+xy+y^2=z^2$ to solve it in natural numbers. The discriminant of it $D=4z^2-3y^2$ and must be perfect square. I wrote Python program to get solutions for $1<x<100$ by ...
yW0K5o's user avatar
  • 367
3 votes
0 answers
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Understanding a Map from rational points to the space of k-rational maps

I am currently working through this paper by Denef, and I have run into a little bit of a snafu with Lemma 3.1. The setup is as follows: let $K$ be any field of characteristic zero, and let $E_0$ be ...
peabody's user avatar
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2 votes
1 answer
213 views

What numbers can be written uniquely as a sum of two squares?

What numbers can be written uniquely as a sum of two squares? I was looking at sequence A125022, which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things ...
huh's user avatar
  • 387
-3 votes
1 answer
130 views

Let $f(x)=x^2-kx$ and $g(x)=x^3-kx$, such that for a rational number $\alpha$, $f(\alpha)$ and $g(\alpha)$ both are rational numbers. Find k [closed]

Let $f(x)=x^2-kx$ and $g(x)=x^3-kx$, $k\in\mathbb{R^+}$ be two real valued functions such that for a rational number $\alpha$, $f(\alpha)$ and $g(\alpha)$ both are rational numbers. Find the range of ...
Maverick's user avatar
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5 votes
0 answers
75 views

Can $\frac{a^n + b^n}{a+b} = c^n$ have a solution? (CONT)

In a question I made few years ago, asking if $\frac{a^n+b^n}{a+b} = c^n$ can have a solution for $a,b,c,n\in \mathbb N$, $a^n+b^n\neq a+b$, and $\gcd(a+b,c)=\gcd(a,b)=1$, the user @MummytheTurkey ...
Juan Moreno's user avatar
  • 1,110
4 votes
1 answer
140 views

Simpler solution of the Diophantine equation $x^6-y^6=z^2$.

In the course of answering another question here (link) I struggled quite hard at a rather unsatisfactory argument to show that $$x^6-y^6=z^2,$$ has no nontrivial integral solutions. For that question ...
Servaes's user avatar
  • 63.4k
5 votes
1 answer
272 views

Find prime numbers satisfying an equation

Find all triplets $(m, n, p)$, where $p$ is a prime number and $m, n ∈ \Bbb N$, such that $p=\frac{m}{4}\sqrt{{2n-m \over 2n+m}}$ My procedure is as follows: $p=\frac{m}{4}{\sqrt{(2n)^2-m^2}\over 2n+m}...
Pranav P J's user avatar
1 vote
1 answer
91 views

Conditions for solutions $a^3 - b^3 = n$

While investigating $a^n - b^n$, I obtained conditions on $n$ for $a^2 - b^2 = n$. However, starting from the 3rd power, the problem becomes significantly more complex. Although something can still be ...
Isid's user avatar
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-2 votes
2 answers
185 views

Are there solutions of $a^n+n+b^n=c^n$ for $n>2$?

This question has been extensively edited to meet site requirements. As is well-known, the Diophantine equation $a^n+b^n=c^n$ has many solutions when $n=2$ (Pythagorean triples) but none when $n>2$ ...
Rajesh Bhowmick 's user avatar
1 vote
1 answer
69 views

solving a Diophantine equation $17+2^m=n^2$ [duplicate]

What is the method to solve $$17+2^m=n^2$$ for positive integers $m$ and $n$. I tried using modular arithmetic which seems did not help.
pipi's user avatar
  • 2,461
1 vote
1 answer
125 views

How do I solve this Diophantine equation?

How do I solve this Diophantine equation, $a^{2n}+b^2=c^{2n}$, where $n$ is any postive integer $>1$ & $a,b,c\ne0$? I tried it by applying the Pythagorean triplets generating formula, but ...
Rajesh Bhowmick 's user avatar
1 vote
1 answer
69 views

Integer solutions for a specific genus one, quartic plane curve

As part of a physics project studying renormalization group flows of scalar field theories, I've come across the following quartic plane curve in the variables $m$ and $n$: $$ 36 + 16 m^4 - 108 n + ...
user2309840's user avatar
1 vote
0 answers
66 views

Doubly exponential diophantine equation

Recently, i've been thinking of the diophantine equation: $A^{B^C}+D^{E^F} = G^{H^I}$ Where, $A, B, C, D, E, F, G, H, I \in [2, \infty)\cap \mathbb{N}$. I discovered that one can construct a solution ...
Bryle Morga's user avatar
1 vote
2 answers
136 views

solutions to the equation $(2m^2-1)^2=2n^2-1$, where $m$, $n$ are positive integers

I'm studying the equation $(2m^2-1)^2=2n^2-1$ ($\ast$), where $m$, $n$ are positive integers. It is known that $m$ can only be 1 or 2 by using Wolfram Alpha. Now I want to prove that result. That is ...
Siyuan Yin's user avatar
0 votes
0 answers
18 views

Approximating an unitary matrix to $\mathbb{Z}[i,1/\sqrt2]$ while keeping unitarity

In this paper (lemma 3) it is claimed that any column of a unitary matrix that has two entries at zero can be approximated to $\mathbb{Z}[i,1/\sqrt2]$ by solving the Diophantine equation $a^2 + b^2 + ...
Cnoob's user avatar
  • 1
1 vote
0 answers
76 views

"Factorization" of the solutions set of a system of linear diophantine equations over non-negative integers

Suppose we have a system of linear diophantine equations over non-negative integers: $$ \left\lbrace\begin{aligned} &Ax=b\\ &x\in \mathbb{Z}^n_{\geq0} \end{aligned}\right. $$ where $A$ is a ...
Alexander's user avatar
1 vote
2 answers
96 views

Solve $3a^2 + 3ab + b^2 - p^3 = 0$ using infinite descent or otherwise?

I'm trying to get my head around infinite descent proofs for Diophantine equations and I was trying to apply it to a problem, and as you can see, I am struggling with it. Consider the identity $$3a^2 +...
DRG's user avatar
  • 367
2 votes
3 answers
133 views

$a^2-2b^2=-73696$

How to find all pairs $(a,b)$ of natural numbers that satisfy $a^2-2b^2=-73696$ and $\gcd(a,b)=4$ The only thing I was able to do is to change the equation to $a_1^2-2b_1^2=-4606$ with $\gcd(a_1,b_1)=...
Mostafa dd's user avatar
-1 votes
2 answers
43 views

Methods for Determining Divisibility by 4 for the Formula $2^n - 46$

I'm working on a problem where I need to determine the conditions under which $2^n - 46$ is divisible by 4, where $n$ is a non-negative integer. I understand that for any power of 2 greater than $2^2$,...
fabul.io's user avatar
3 votes
2 answers
153 views

Finding integer solutions for a logarithmic equation with constraints

I am struggling with a logarithmic equation where I am required to find integer solutions for a such that x is also an integer ...
fabul.io's user avatar
-3 votes
1 answer
112 views

Solving diophantine equations: Finding integer solutions: $(x+1)^2+(x+2)^2+...+(x+2001)^2=y^2$ [closed]

I would like to find all integer solutions to the Diophantine equation. I wonder if there is a way to solve it using Fermat. $(x+1)^2+(x+2)^2+\ldots+(x+2001)^2=y^2$
An Nguyen's user avatar
5 votes
1 answer
229 views

Find all $(a,b)\in\Bbb{N},$ such that $5^a +2^b +8$ is a perfect square.

Find all $(a,b)\in\Bbb{N},$ such that $5^a +2^b +8$ is a perfect square. My approach: Let $5^a +2^b +8=k^2\implies \bigg(k+5^{\frac{a}{2}}\bigg)\bigg(k-5^{\frac{a}{2}}\bigg)=8\bigg(1+2^{b-3}\bigg).$ ...
Vulch's user avatar
  • 61
1 vote
1 answer
75 views

Generalized Pell equation $ x^2 - (k^2-1)y^2 = p $ and solutions recurrence relations

Solving the Pell equation $ x^2 - (k^2-1)y^2 = 1 $, the general solutions for $y$ are generated by the recurrence relation $y_{n+2} = 2k\cdot y_{n+1} - y_n, y_0 = 0, y_1 = 1$ which is the same of the ...
user967210's user avatar
3 votes
1 answer
252 views

On why solutions to $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$ also come in pairs

Jacobi and Madden found infinitely many primitive solutions to, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$ using an elliptic curve. We will use a different approach that, like the method for, $$a^4+b^4+c^4 = d^...
Tito Piezas III's user avatar
1 vote
1 answer
65 views

Finding Integer Solutions for a Set of Equations Involving Powers and Logarithms

I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as ...
fabul.io's user avatar
5 votes
1 answer
126 views

Why is the number 3 come up so often in Chaos theory and Undecidability as a boundry? Is it just a coincedence?

What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order. Examples: Quadratic Diophantine equations are always ...
Colonizor48's user avatar
1 vote
1 answer
64 views

Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?

The questions asks: Define a formula to yield the Kth N for which there exists an integer X less than or equal to N for which the sum of the integers from 0 to X (inclusive) is equal to the sum of the ...
BlueInfinite1729's user avatar
1 vote
2 answers
175 views

Solve in $\mathbb{N}$ the equation $3x^2 + 2x = y^2$.

Solve in $\mathbb{N}$ the equation $3x^2 + 2x = y^2$. After factoring, we get $x(3x+2)=y^2$. If LHS is a perfect square and we let $x = n^2z$, where $z$ is not divisible by the square of any prime ...
math.enthusiast9's user avatar
8 votes
3 answers
777 views

On why solutions to $x^4+y^4+z^4 = 1$ come in pairs

I. Question We have a quartic in $v$ to be made a square, $D^2 = 4(-6 - 2u + u^2)(2 - 2u + u^2) - 8(-2 - 4u + u^2)(2 - 2u + u^2)v - 16u(4 - 3u + u^2)v^2 - 4(4 - 12u + 4u^2 - 2u^3 + u^4)v^3 + (4 - 8u - ...
Tito Piezas III's user avatar
0 votes
1 answer
46 views

Solutions of Diophantine Equations with Restricted Terms

For the equation $x_1+x_2+\dots+x_n=k$, I know that the total number of nonnegative solutions is ${k+n-1 \choose k}$. My question is how does the number of solutions change if all $x_i$ have to be ...
frog3549's user avatar
2 votes
1 answer
91 views

Solution of non linear homogeneous diophantine equations

I was given the following determinant to evaluate. \begin{pmatrix}x^3+1&x^2y&x^2z\\ xy^2&y^3+1&y^2z\\ xz^2&yz^2&z^3+1\\ \end{pmatrix}. I simplified it to $$x^3+y^3+z^3+1$$ In ...
Aarush Saharan's user avatar
1 vote
1 answer
142 views

Two grasshoppers jump in opposite directions - find effective solution?

I took a test in number theory and modular arithmetic today, and one of the questions looked like this: Two grasshoppers A and B hop in a circle that is divided into 81 squares. The squares are ...
naytte2's user avatar
  • 454
4 votes
1 answer
155 views

On counterexamples to Euler's conjecture using Bremner's and Durman's elliptic curves

Noam Elkies found the first counter-example to Euler's sum of powers conjecture that, $$a^4+b^4+c^4 = d^4$$ was not solvable by expressing the equation as an intersection of two quadric surfaces ...
Tito Piezas III's user avatar
6 votes
3 answers
404 views

Still more elliptic curves for $a^4+b^4+c^4=d^4$

There are 31 known primitive solutions to $a^4+b^4+c^4 = d^4$ with $d<10^{28}.$ (Update: As of Feb. 21, there are now 93. See this MSE table.) Old statistics are, \begin{array}{|c|c|} \hline \text{...
Tito Piezas III's user avatar
2 votes
0 answers
86 views

Robbins Pentagons with Irrational Diagonals

A Robbins pentagon is a cyclic pentagon whose edge lengths and area as all rational numbers. All known Robbins pentagons have rational diagonals, but it has not yet been proven that no Robbins ...
Michael Ejercito's user avatar
2 votes
2 answers
147 views

Generalized 4-variable Diophantine equation

The diophantine equation $(a^2+b^2+c^2+d^2)^2 = 2(a^4+b^4+c^4+d^4)$, which it's the case of Descartes' circle theorem where all curvature are integer perfect squares, was cleverly solved by Euler ...
Falcon's user avatar
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