1
vote
0answers
46 views

Diophantine Equation $2x^2+25=y^3$

I'm trying to find integer solutions to: $2x^2+25=y^3$. Here's what I've managed to do so far: y is odd. y and x are co-prime. In $\mathbb{Q}(\sqrt{2},i)$ we can write: ...
0
votes
1answer
47 views

$(x+\frac{1}{x})(y+\frac{1}{y})$ is equal to positive integer, solutions. [closed]

$(x+\frac{1}{x})(y+\frac{1}{y})$ is equal to positive integer. General proof/(conditions?) for positive the solution.
1
vote
1answer
100 views

$x^x + y^x=x^y + y^y$ positive integer solutions?

required is positive solutions for $x^x + y^x=x^y + y^y$? And negative integer solutions as well if possible?
0
votes
2answers
32 views

For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$.

For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$. $(i) (a,b) = (3,7)$ $(ii) (a,b) = (5,7)$ $(iii) (a,b) = (4,11)$ ...
0
votes
1answer
63 views

Solutions of Diophantic Equation

Given the Diophantic Equation $$1188x +63y =26$$ Task: Find integer solution(s) I found that $$1188x +63y =26 \Longleftrightarrow 132x+7y = \frac{26}{9}$$ One can easily see that LHS ...
6
votes
2answers
103 views

Equation: $(x^2-9y^2)^2=33y+16$

I want to know the solution of the equation $(x^2-9y^2)^2=33y+16$ in positive integers. I know it has solution $(\pm2;0)$ but I can't prove that it doesn't have other solutions. Please help.
1
vote
1answer
59 views

Need a proof to show all the units are satisfied $\mathbf{Z}\sqrt{2}$ is the all the integer solution in Pell equation [duplicate]

We know the integer solutions of Pell's equation $$a^2-2b^2=\pm1$$ correspond to the units of $\textbf{Z}[\sqrt{2}]$. How can we prove this?
1
vote
0answers
87 views

Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$

All the units are satisfied Pell's equation $a^2-2b^2=\pm1$ for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$. Here is my proof: Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies ...
0
votes
2answers
476 views

Solving linear Diophantine equations in 3 variables

I have to solve a linear Diophantine equation in more than 2 variables> I sort of have an idea of how to solve it, but I'm not clear how. One of the problems is this: $$12x + 21y +9z + 15w = 9$$ How ...
3
votes
0answers
94 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
3
votes
1answer
157 views

Find all positive integers m, n, p such that $(m+n)(mn+1)=2^p$

Find all positive integers m, n, p such that $$(m+n)(mn+1)=2^p$$ Please give me some hints Thanks
1
vote
0answers
58 views

How to solve the diophantine equation:$ xa^3+yb^3=c^3$

Let $a,b,c,x,y \in \mathbb{Z}> 1$. Any hint on how to solve of the diophantine equation $ xa^3+yb^3=c^3$?
7
votes
3answers
208 views

Prove that there are exactly 16 solutions to this problem.

Show that are are only 16 integer solutions to the following equation: $$11x + 8y + 17 = xy$$ What I tried: I took a modulo 2, and I got that $y$ must be even and $x$ must be odd. But beyond that, I ...
5
votes
0answers
171 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
2
votes
1answer
113 views

Total no. of ordered pairs $(x,y)$ in $x^2-y^2=2013$

Total no. of ordered pairs $(x,y)$ which satisfy $x^2-y^2=2013$ My try:: $(x-y).(x+y) = 3 \times 11 \times 61$ If we Calculate for positive integers Then $(x-y).(x+y)=1.2013 = 3 .671=11.183=61.33$ ...
4
votes
1answer
150 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
0
votes
3answers
85 views

Finding K value to solve a problem using diophantine equation

I have to prove that any number that divided by 5 gives a remainder of 1 and divided by 7 a remainder of 2, also gives a remainder of 16 when divided by 35, using a diophantine equation. So first I ...
2
votes
1answer
132 views

Solutions of Diophantine equations in Natural numbers

The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$ of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail ...
1
vote
5answers
170 views

Solve $2a + 5b = 20$

Is this equation solvable? It seems like you should be able to get a right number! If this is solvable can you tell me step by step on how you solved it. $$\begin{align} {2a + 5b} & = {20} ...
3
votes
2answers
292 views

Modification of 5th question from BMO'81

First of all I will introduce original problem (Question 5 from British Mathematical Olympiad). You can find complete list of BMO'81 there BMO'81. Find, with proof, the smallest possible value ...
5
votes
5answers
2k views

Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$

I'm working on a homework problem that is as follows: Suppose that $n$ is a positive even integer with $n/2$ odd. Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$. ...
3
votes
3answers
223 views

Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points

Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve. By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$. By implicit ...
8
votes
1answer
289 views

Find all positive integers $L$, $M$, $N$ such that $L^2 + M^2 = \sqrt{ N^2 +21}$

Sorry, this is very much 'can you do my homework' but I have a little competition at work that requires me to solve (and prove) the following. Find all positive integers $L$, $M$, $N$ such that ...
2
votes
2answers
2k views

polynomial and integer roots

I'm doing some homework for a computer science class. It's been so long since I've done math, I have a question that assumes math knowledge that confuses me. Given: Whether a diophantine polynomial ...
12
votes
4answers
725 views

Proof that $x^2+4xy+y^2=1$ has infinitely many integer solutions

The question would, naturally, be very straight forward if there was a $2xy$ instead of a $4xy$. Then it would simply be a matter of doing: $$ x^2+2xy+y^2=1\\ (x+y)^2=1\\ \sqrt{(x+y)^2}=\sqrt{1}\\ ...
0
votes
1answer
69 views

Linear Diophantine Set Proof

Let's say I have set S and T being the set of all integer solutions to $ax+by=c$ and $ax+by=nc$ respectively, and set S* might be the same as set T. S* = $\{ (n x_0 + n y_0) | (x_0, y_0) \in S\}$ ...
4
votes
1answer
127 views

A quartic diophantine equation

Here is the statement: Let $a,b \in \mathbb{Z}$ positive integers such that $a^2=b^4+b^3+b^2+b+1.$ Prove $b=3.$ I've tried is the following: Let $\Sigma=b^4+b^3+b^2+b+1$. If $a\equiv 0\mod 3$, then ...
4
votes
3answers
1k views

$\mathbb Z[\sqrt 3]$ contains infinitely many units

I'm asked to show that there are infinitely many units in the ring $\mathbb Z[\sqrt 3]$. But I don't really see a good approach to this one, so far. Some thoughts: The inverse of $a+\sqrt3 b$ ...
3
votes
3answers
102 views

Algebra Question System of Equations

How would one go about solving the system of five equations: $p^2=p+q-2r+2s+t-8$, $q^2=-p-2q-r+2s+2t-6$, $r^2=3p+2q+r+2s+2t-31$, $s^2=2p+q+r+2s+2t-2$, $t^2=p+2q+3r+2s+t-8$ over the integers? ...
4
votes
2answers
235 views

Sum of digits algebra problem

How would one go about finding all 3-digit positive integers $ \overline{abc}$ with the property $\overline{abc}=abc(a+b+c)$, where $ \overline{abc}$ would be the decimal representation of a number. ...
5
votes
1answer
139 views

Diophantine equation : $N= \frac{x^2+y}{x+y^2}$

I am looking for information about the following diophantine equation : $N = \displaystyle\frac{x^2+y}{x+y^2}$ Has it been studied ? Is there any efficient algorithm to solve it? Any links? I ...
0
votes
1answer
111 views

Linear diophantine equation

a) $$ 130x + 143y = 5957 $$ b) $$ 44x + 19y = 75 $$ the theorem says ax + by = c has solution if and only if d | c however I work out both question with no solution as a) $$ 143 = 130 . 1 + 13 ...
2
votes
2answers
497 views

Show that $x^2 - 3y^2 = n$ either has no solutions or infinitely many solutions

I have a question that I have problem with in number theory about Diophantine,and Pell's equations. Any help is appreciated! We suppose $n$ is a fixed non-zero integer, and suppose that $x^2_0 - 3 ...
1
vote
2answers
382 views

$x^2 + 2 = 5y$ ($x$ and $y$ positive integers)

Question: Determine all positive integers $x$ and $y$ that satisfy the equation $x^2 + 2 = 5y$.
1
vote
1answer
256 views

Odd positive integers that satisfy $a^2 - b^3 = 4$

Are there any odd positive numbers that satisfy the equation: $a^2 - b^3 = 4$ ? I am certain that there are none but can't prove it. How would you prove that?