Questions tagged [diophantine-equations]

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

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25 views

A Number Theory problem: solving $p^2=n^5+1$ [closed]

Solve for all integers that satisfy this equation $p^2=n^5+1$.
2
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1answer
30 views

Find all integer solutions for the equation [duplicate]

How to find all integer solutions for the equation $y = \frac{a+bx}{b-x}$, where a and b are known integer values? P.S. x and y must be integer at the same time
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31 views

On the composition of binary quadratic forms (esp. Gauss Disquitiones Arithmeticae Art. 235/236)

In Composition of Binary Forms and the Foundation of Mathematics, Harold M. Edwards says that Gauss [cf. Art 235/236 of Disquisitiones Arithmeticae] proved the following result. Theorem. Let $f$ and $\...
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57 views

When is $8^n b^n + \cdots + 8 b + 1$ a perfect square (for positive integer $b$)? [duplicate]

I watched the video How perfect are these squares?? - YouTube by Michael Penn, where there's a question in the thumbnail which isn't answered. The thumbnail question is: When is $$2^n b^n + \cdots + ...
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1answer
53 views

What are the integer solutions to $\cos(\sqrt{n^2-1}) = \frac{1}{n}$?

$$\cos(\sqrt{n^2-1}) = \frac{1}{n}$$ I was wondering if there existed integer solutions to to this equation apart from n=1. I've thought that there are probably no more solutions, because RHS is ...
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72 views

Perfect Square With Two Integer Variables

I am trying to solve a number theory problem in general form. However, I got stuck in the following step: $a,b,n \in \mathbb Z^{+}$ for which values of $n$, this equation is solvable $\frac{(n+1)(n+2a)...
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38 views

How to approxmate a real number by a sum of two real with integer weights?

Given three real numbers, $a$, $b$, and $c$, I am wondering how to approximate $c$ by the sum $an+bm$ for integers $n$ and $m$ up to the desired accuracy. More precisely, define $z:\mathbb{Z}\times \...
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1answer
62 views

Prove that $y^2z^2 - y^2 -z^2$ is not a perfect square for any $y,z \in \mathbf{N}$ [duplicate]

So, I was looking at this problem which asks to show that $x^2 + y^2 + z^2 = 2xyz $ where $ x,y,z \in \mathbf{N}$ has no solution. Viewing the equation as quadratic in $x$ and solving for $x$ gives $$ ...
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Can we find integers $X$ and $Y$ such that $(a + b)X + (a^2 - ab + b^2)Y = c(ax + by)$ where all variables are integers?

Here is a simpler problem to show what I am trying to solve: Say if we are asked a different but similar problem like this: Find integers $X$ and $Y$ such that $(a + b)X + (a - b)Y = c(ax + by)$ where ...
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Project on Diophantine equations [closed]

My favourite courses are on number theory, ring theory and algebraic geometry. Since it seems to me that diophantine equations live in the intersection of these three areas of mathematics, I was ...
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60 views

Find all linear relations of the following numbers in $\mathbb{Z}$

I have to find all linear relations of the numbers $3,4$ and $5$ in $\mathbb{Z}$ and find if there are relations between these relations. Let $x=3$, $y=4$ and $z=5$. As far as I understood it, i ...
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1answer
143 views

Solve in positive integers the equation $x^3 + y^3 + z^3 - 3xyz = 1517$

Solve in positive integers the equation $$x^3 +y^3 +z^3 −3xyz = 1517$$ Initially, I tried to factorize the LHS and my last step was $$(x+y+z)(x^2+ y^2 + z^2 -xy -zx - yz) = 37×41$$ How do I find the ...
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75 views

Diophantine with factorials

This is a problem I encountered on a competition Discord server, apparently, there is an elementary solution, but I'd honestly be fine with any solution. Wolfram Alpha solves the problem, here's the ...
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1answer
37 views

How to solve Linear diophantine equation ax+by=4ab

$ Answer : x= 3b, y=a $ $ x =2b,y=2a $ $ x= b, y=3a $ $. $ $ ax+by=4ab $ $ (0,4a) $ $ x=x' $ $ y=y'+4a $ $ ax'+b(y'+4a)=4ab $ $ ax'+by'+4ab=4ab $ $ ax'+by'=0 $ $ ax'=-by' $ $ x':y' = (- b) :...
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2answers
68 views

How to solve this diophantine equation: $x^2+2y^2=x^2y^2-2000$

Solve the following diophantine equation: $$x^2+2y^2=x^2y^2-2000$$ I tried this by adding and subtracting terms, but so far, no avail. $(\pm 2y^2+4xy, \pm2xy)$ I don't know how to start either. ...
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2answers
70 views

Are these the only “intersections” between the following series, and why all of them are multiples of $10$

As a web developer that programs in PhP, I enjoy running some scripts to see some of math wonders, however, PhP is limited to large calculations. I wanted to see if there are any intersections for the ...
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2answers
64 views

Solution of the Diophantine equation

What are the possible triples (x,y,z) in positive integers such >that, $$(x+y)^{2}+3x+y+1=z^{2}$$ I have used the inequality approach and many others but wasn't able to find an answer.
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1answer
47 views

Ramanujan-Nagell-ish diophantine equation

The task is to find all $a, b \in \mathbb{Z}_+$ s.t.: $$2^a+17=b^4$$ I tried reducing modulo 17, but it doesn't really give much. Also a 4th power can $\equiv17$ for any $a$ big enough. Computer gives ...
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105 views

Positive integer solutions to $pxy+x+y=p\#$

Let $p$ be prime and $p{\#}$ the product of all primes not larger than $p$. Are there any positive integers $x$ and $y$ such that $pxy+x+y=p{\#}$. It appears there are no solutions. There are no ...
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61 views

Positive integer solutions to $y^2=a(1+xy-x^2)$

Let $a>3$ be an integer. Define a sequence $X$ as : \begin{equation} \begin{aligned} x_1 & = 1\\ x_2 & = a-1\\ x_n & = (a-2)x_{n-1}-x_{n-2}, \ \ n\ge3 \end{aligned} \...
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Reference request: there are only two integer solutions to $2^{2a} + 3^{2b} = 5^c$. [duplicate]

I believe there are only two non-negative integer solutions to $$2^{2a} + 3^{2b} = 5^c.$$ The solutions I have are $a=1,b=0,c=1$ and $a=2,b=1,c=2$. I'm not certain this is correct. I'd like to know if ...
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Can you prove mu-recursive functions are Diophantine without bounded universal quantifiers?

Most proofs I’ve come across for the unsolvability of Hilbert’s tenth problem show that every recursive function is Diophantine using the approach of mu-recursive functions, i.e. they show that the ...
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1answer
64 views

Find all integer solutions to $n^2 - n = c\cdot 2^{J+1}$

I am trying to construct a matrix with dimensions such that the number of unique off-diagonal elements (i.e. the number of elements in the upper or lower triangle) is proportional to a power of 2. By ...
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32 views

How to solve this Diophantine equation: the commensuration lattice of two honeycomb lattices with different periods

I am trying to solve the commensuration lattice (lattice vector) of two honeycomb lattices with different periods, which leads to the the following Diophantine equation. So given real number $\theta$ ...
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2answers
185 views

Uses of Vieta Jumping in research mathematics?

Vieta jumping has been a prominent method for solving Diophantine equations since 1988. It was popularized when it was used to solve an IMO problem, but has it been applied to research mathematics, ...
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42 views

Are there coprime nonzero integers $u,v,w$ such that $2u(u^2+pv^2)=w^p$ where $p>3$?

Let $p$ be a prime. Let $u,v$ be coprime non zero integers,$w$ is an integer. Does the equation $$2u(u^2+pv^2)=w^p$$ always yield an infinite descent argument? For some primes like (3,7), it is ...
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1answer
123 views

$\frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc}$ and $a^2 + b^2 = c^2$

I have found this in an Romanian magazine. We have to solve for natural numbers: $$\frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc}$$ $$a ^ 2 + b ^ 2 = c ^ 2$$ After some elementary ...
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0answers
45 views

Finding Lattice Points on a Sphere

I want to find all the lattice points on a sphere with integer radius, or equivalently all solutions to the diophantine equation $x^2+y^2+z^2=r^2$. We can see from oeis that the number of solutions, ...
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3answers
98 views

Solve for integers $x, y$ and $z$: $x^2 + y^2 = z^3.$

Solve for integers $x, y$ and $z$: $x^2 + y^2 = z^3.$ I tried manipulating by adding and subtracting $2xy$ , but it didn't give me any other information, except the fact that $z^3 - 2xy$ and $z^3+2xy$...
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1answer
76 views

Diophantine equation $3x^2+y^2=z$ [closed]

I am currently facing a Diophantine equation $3x^2+y^2=z$, in which $x$, $y$, $z$ are integers. My major is not math and I am entirely new to Diophantine equation. I googled this but only found ...
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1answer
33 views

Why does $p \nmid m$ imply $\left(\frac{-4m^2} p\right) = \left(\frac{-1 } p\right)$?

$\newcommand{\Leg}[2]{\left(\frac{#1}{#2}\right)} $ Note: In this question $ a \mid b $ denotes a divides b and $\Leg a b$ denotes Legendre's symbol. Theorem 9.12 in Introduction to Analytic Number ...
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1answer
67 views

Integral right and isosceles triangles with equal area and perimeter

There are many different tasks in pictures on the Internet. I found one picture and it interested me. And two questions. The first. Did I write this system correctly? That is, to reformulate the ...
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1answer
62 views

Finding the solutions of an equation in positive intgers

We are asked to find all positive integer solutions of the equation $$x^7+7=y^2$$ or to show that it does not have any solutions. It is clear that $x$ can not be even. So $x$ is odd and $y$ is even (...
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3answers
148 views

Find the integer solution of the equation $x^3+y^3=x^2+y^2+42xy$

Find the integer solution of the equation $x^3+y^3=x^2+y^2+42xy$ I try $x=0$ We have: $y^3-y^2=0 \Longrightarrow \left\{\begin{array}{l} y=0 \\ y=1 \end{array}\right.$ I think, this equation only $(x,...
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0answers
18 views

Diophantine equation where constants are -1, 0, or 1

Consider the Diophantine equation $$\sum_{I \in \{0, 1\}^n} a_I x^I y^{1 - I} = 0,$$ where $a_I \in \{-1, 0, +1\}$ are the known constants, $a_{0, \dots, 0}= -1$, $x \in \mathbb{Z}^n$ and $y \in \...
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1answer
63 views

Determine the smallest $n$ such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ has as positive integer solutions exactly $15$ pairs

Determine the smallest natural number $n$ such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ has as solutions exactly $15$ ordered pairs of natural $(x,y)$. I found out that this problem is not an ...
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1answer
29 views

Suppose the equation $ax+by=c$ has $m$ positive solutions. How many positive solutions does the equation $ax+by=c+ab$ have?

Suppose that $a,b,c$ are positive integers. Suppose the equation $ax+by=c$ has $m$ positive solutions. How many positive solutions does the equation $ax+by=c+ab$ have? I know $ax+by=ab$ does not have ...
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0answers
587 views

About $a^3+b^3=c^3$. - Could you please review this kindly? [closed]

Proof: $a^3+b^3=c^3$ has no solution in whole numbers. The proof was already given by Euler; Euler used complex algebra and completly different calculations. This proof uses real number algebra and ...
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113 views

Gcd of two numbers of the form $x^a-x^b$

Inspired by this question, which noted that for all natural numbers $a>2$, $(2^{15}-2^3)|(a^{15}-a^3)$. My question deals with generalizing this: Let let $a,b$ be integers such that $a>b\geq1$. ...
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2answers
59 views

$\frac{ (359\cdot (109+215\cdot x)-1)}{10^x}=y$

Consider the diophantine equation: $\frac{ (359\cdot (109+215\cdot x)-1)}{10^x}=y$, for x,y positive. The only solution I found is $x=2$, $y=1935$. Can it be proven that if there is a solution there ...
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4answers
68 views

Prove there are no integer solutions for $x^2 + 3y^2 = 8$

I know there are analogous questions to mine, but I'm seeking a different approach. All the solutions I saw use some sort of modular arithmetic trick, but I'd like to prove this without resorting to ...
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0answers
114 views

Find a positive integer $i$ such that $9i + 1$ divides $2 \times 10^i - 1$

I have written a Python program running over $i$, but up to billions there is no solution, so I guess there is no solution. Trying to prove that, I looked at multiplicative order, but I do not get a ...
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0answers
24 views

Are there integers nonzero integers $a,b,c,d,x,y,m,n,p$ such that $(a^2-mb^2)(c^2-nd^2)=x^2-py^2$? ($m,n,p)$ are square free non equal integers.

We are all familiar with Fibonacci-Brahmagupta's identity: $$(a^2-mb^2)(c^2-md^2)=(ac+ mbd)^2-m(ad+bc)^2$$ I am trying to find whether there is a similar identity: $$(a^2-mb^2)(c^2-nd^2)=x^2-py^2$$ ...
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3answers
42 views

On difference of two sums of two squares plus difference of two cubes [closed]

Is there a way to show that every integer $n$ can be represented by $(a^3+b^2+c^2)-(d^3+e^2+f^2)$ Where $a,b,c,d,e,f$ are all integers?
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44 views

Approximating the number of lattice points on a sphere

When deriving the Rayleigh-Jeans law in physics, one must count the number of solutions $(a,b,c)$ to the Diophantine equation $$a^2+b^2+c^2=R^2.$$ The source I have linked approximates the number of ...
5
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1answer
143 views

A problem with sequences

I have a problem I don't seem to be able to solve other than by brute force. Consider the increasing sequences of $n$ positive integer numbers such that all the $n−1$ differences between any two ...
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0answers
128 views

find all the integral solution for $y^2 + 31 = x^3$

Find all the integral solution for $y^2 + 31 = x^3$. I am reading Ireland's book 'a classical introduction to modern number theory', this is one of the exercises. The hint behind the book is $y^2 + 4 =...
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2answers
189 views

Diophantine equation $2^x-3^y=2021$ [closed]

$$2^x-3^y=2021$$ where $x,y$ are non-negative integers. I only found $2^{11}-3^3=2021$.
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52 views

Review on diophantine equations

Solving diophantine equations was a classical motivation for development of mathematics. As far as I know methods used there are rather diverse: at least some hard arithmetic geometry and diophantine ...
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1answer
78 views

Could it be generally proved that some positive integer is not of the form $x+y+2xy$? [closed]

I believe that the answer to the question is negative, because if it were possible, we would be generally able to tell if some positive integer is prime or not using the sieve of Sundaram, but I would ...

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