# Questions tagged [diophantine-equations]

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

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

Solve for all integers that satisfy this equation $p^2=n^5+1$.
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc}$ 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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### 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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### 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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### 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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### Why does $p \nmid m$ imply $\left(\frac{-4m^2} p\right) = \left(\frac{-1 } p\right)$?

$\newcommand{\Leg}{\left(\frac{#1}{#2}\right)}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\frac{ (359\cdot (109+215\cdot x)-1)}{10^x}=y$

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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### 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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### 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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### 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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### 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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### 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 ...
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 ...