Questions on finding integer/rational solutions of equations.

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2
votes
2answers
243 views

integer solutions of $xy+9(x+y)=2006$

How many integer solutions does $xy+9(x+y)=2006$ have ? Here x and y are both integers . My trying : I have tried to solve this problem But I have no idea to solve this . Please help
1
vote
1answer
136 views

How to prove that $x^2 - 3y^2 = 11$ is not possible as long as x and y are integers.

Really No idea how I can go about solving it. What I did was, $$y = \sqrt{\frac{x^2 - 11}{3}}$$ but cant go beyond that. Can anybody help?
0
votes
2answers
154 views

Linear diophantine equation: count number of solutions that satisfies a certain constraint

Given a linear diophantine equation like $$\sum_{i=1}^na_ix_i=b$$ with $x_i$ as variables and $a_i$ and $b$ as fixed integer parameters, I want to count the number of solutions that satisfies the ...
3
votes
1answer
473 views

Proving that diophantine equation has no solutions

I am trying to show that the equation $x^5y + 5x^3 - xy^5 = 1$ has no solutions. Anyone has an idea on this?
6
votes
5answers
933 views

Proof that the equation $x^2 - 3y^2 = 1$ has infinite solutions for $x$ and $y$ being integers

I have seen the Pell's equation wiki page but I need to prove this from scratch without mentioning any formula. I have also seen multiple answers on this site but the answers tend to skip over and ...
0
votes
2answers
71 views

Diophantine equation without unique formula for solutions

Every one know solutions of the Diophatine equation $x^2+y^2=z^2$ which are given by formula $x=t(a^2-b^2)$, $y=t(2ab)$ and $z=t(a^2+b^2)$. In this exemple one proove that all the solutions are in ...
6
votes
2answers
119 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.
0
votes
1answer
78 views

How to solve a system of equations with only integers

$a+2y = 320$ $2b+3y = 320$ $3c+4y = 320$ And a, b, c, and y are all integers. How do I find all possible solutions, if any?
0
votes
2answers
74 views

If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $n^3-32n^2+n=k^2$.

If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $$n^3-32n^2+n=k^2$$ I've tried checking congruence modulo $4$. I have that n is divisible by 4. Let $n=4u$, where $u\in\mathbb N$. Then the equation ...
0
votes
1answer
106 views

Diophatine equation $x^2+y^2+z^2=t^2$

Probably duplicate but I don't find: I'd like to solve the diophantine equation $$x^2+y^2+z^2=a^2$$ which has solutions, by exemple $1^2+2^2+2^2=3^2$ or $2^2+3^2+6^2=7^2$. Every such solution gives ...
11
votes
0answers
388 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
0
votes
2answers
110 views

Proof that $x^2 -3y^2 = 1$ has infinite solutions. ( x and y are integers)

I have to explain this to my brother who is in eighth grade and I would really love if you could tell this in simple terms (I'm no Maths guy).
8
votes
1answer
286 views

If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$. [duplicate]

If $p,q$ are prime, solve $$p^3-q^5=(p+q)^2$$ I can't think of a nice idea for the solution. Since there's a solution $(7;3)$, consisting of two distinct numbers, I really doubt modular arithmetic ...
3
votes
2answers
85 views

If $x,y\in\mathbb Z$, solve $x^2+xy+y^2=x^2y^2$.

If $x,y\in\mathbb Z$, solve $$x^2+xy+y^2=x^2y^2$$ We could try some factorizations. $x^2+y^2=xy(xy-1)$. We may as well add $2xy$ to both sides: $(x+y)^2=xy(xy+1)$. Then we could subtract $x^2y^2$ ...
0
votes
1answer
200 views

Is $x^3 + y^3 = z^3$ possible?

Is $x^3 + y^3 = z^3$ possible when $x$, $y$ and $z$ are integers? If not, how to prove that they are not possible? (I am a grade 10 student so please answer in a simple way)
1
vote
0answers
28 views

Is there an analogue of the 15 theorem for cubic forms?

The 15 theorem states that if an integral quadratic form with integral matrix represents the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15, then it represents all numbers. Is there an analogue of this theorem ...
4
votes
1answer
92 views

Find all $x,y\in\mathbb{Z}$ s.t $2x^3-7y^3=3$

Find all $$x,y\in\mathbb{Z}$$ such that $$2x^3-7y^3=3$$ Solution: We consider first $$2x^3-7y^3\equiv3 \pmod 2$$ $$5y^3\equiv 1 \pmod 2$$ $$y^3\equiv 1 \pmod2$$ which has solution $y\equiv 1 ...
4
votes
3answers
161 views

Discrete math. Solve the equation in the set of natural numbers.

I have to solve the equation $$m^4-n^4=5(m^3+n^3)$$ in the set of natural numbers. I wrote a simple code in java and i solved the equation. Only solution in the set of natural numbers is $m = 6$ and ...
9
votes
1answer
119 views

Find all of the integer solutions of $x^3y+y^3z+z^3x=0.$

Using Fermat's Last Theorem, find all of the integer solutions of $x^3y+y^3z+z^3x=0.$ I try to make some substiution so as to transform the equation into a form like a fermat equation but in vain, ...
2
votes
1answer
107 views

Diophantine quartic equation in four variables, part deux

A recent Question asked for all positive integer solutions of a simple quartic in four unknowns: $$ wxyz = (w+x+y+z)^2 \tag{1}$$ whose satisfaction is necessary for the integer side lengths ...
0
votes
1answer
70 views

Using existential quantifiers to turn equalities into inequalities

I have a formula of the form $f(x)^2 = 0$, where $x \in \mathbb{R}^n$ and $f$ is a Diophantine polynomial. I am wondering if there is a general way to produce a formula of the form $\exists y \in ...
7
votes
1answer
165 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...
3
votes
6answers
624 views

Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
3
votes
1answer
194 views

Solve $7x^3+2=y^3$ over integers

I need to solve the following solve $7 x^3 + 2 = y^3$ over integers. How can I do that?
3
votes
1answer
166 views

Subsets of all Diophantine's sets

Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable $\Leftrightarrow$ graph of function is Diophantine. Consider some subset $S$ of computable functions (for example some Grzegorczyk's class or ...
3
votes
2answers
199 views

A partition of tenth powers into ten parts with equal sums

Is there a natural number n for which the numbers $1^{10},2^{10},3^{10}\ldots, n^{10}$ we can put into 10 groups, such that the sum of the numbers in each group is the same?
1
vote
1answer
155 views

Stuck solving an equation using the floor operator.

I am not entirely familiar with the equation ninja'ing involving the floor operators. Here is my problem. I need to solve for $x$. Everything is an integer, including $x$: $$ a - 1 = \lfloor {\frac{x ...
5
votes
4answers
272 views

Find all integer solutions: $x^4+x^3+x^2+x=y^2$

Find all integer solutions of the following equation: $$x^4+x^3+x^2+x=y^2$$
2
votes
1answer
185 views

Solve in $\mathbb Z$ the equation: $x^5 +15xy + y^5=1$

Solve in $\mathbb Z$ the equation: $x^5 +15xy + y^5=1$ I tried: $x(15y+x^4)+y^5=1$ But don't have much ideas on how to continue, thanks!
4
votes
5answers
214 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
5
votes
3answers
464 views

$(1+1/x)(1+1/y)(1+1/z) = 3$ Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers.

Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers and $$(1+1/x)(1+1/y)(1+1/z) = 3.$$ I know $(x+1)(y+1)(z+1) = 3xyz$ which is no big deal. I can't move ...
9
votes
0answers
266 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
2
votes
1answer
84 views

If $p$ is prime and $p>3$ and $k,l,m,n,p\in\mathbb N$ and $p^k+p^l+p^m=n^2$, prove that $8\mid p+1$.

If $k,l,m,n,p\in\mathbb N$ and a prime number $p>3$ that satisfies $$p^k+p^l+p^m=n^2$$ is chosen, prove that $8\mid p+1$. $n^2$, when divided by $8$, gives a remainder $1$ (it can't give the ...
5
votes
1answer
183 views

Diophantine Equation in $\mathbb{Z}$

I would like to know how to solve $2x^2 - y^{14} = 1$ in integers. I've transformed it into $(y^7 - 1)^2 + (y^7 + 1)^2 = (2x)^2$ and I have stopped here.
2
votes
3answers
157 views

Proof that $x^2 - 2y^2 = -1$ has a recurring solution for $x$

I read here about the following variation on Pell's equation: $$ x^2 - 2y^2 = -1.$$ According to Dario Alpern's solver, the equation has infinite integer ...
7
votes
5answers
196 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
5
votes
4answers
2k views

Find all solutions of $1/x+1/y+1/z=1$, where $x$, $y$ and $z$ are positive integers

Find all solutions of $1/x+1/y+1/z=1$ , where $x,y,z$ are positive integers. Found ten solutions $(x,y,z)$ as ${(3,3,3),(2,4,4),(4,2,4),(4,4,2),(2,3,6),(2,6,3),(3,6,2),(3,2,6),(6,2,3),(6,3,2)}$. ...
2
votes
4answers
365 views

System of quadratic Diophantine equations

Is there a method for determining if a system of quadratic diophantine equations has any solutions? My specific example (which comes from this question) is: $$\frac{4}{3}x^2 + \frac{4}{3}x + 1 = ...
5
votes
2answers
143 views

Sequential sums $1+2+\cdots+N$ that are squares [duplicate]

While playing with sums $S_n = 1+\cdots+n$ of integers, I have just come across some "mathematical magic" I have no explanation and no proof for. Maybe you can give me some comments on this: I had ...
1
vote
3answers
205 views

Linear recurrence solution to Diophantine equation

I have a Diophantine equation of the form: $$ax^2 + bx + c = y^2, \quad x, y \in \mathbb{Z^+}$$ Is it true that there will always be a linear recurrence formula that generates all the solutions for ...
2
votes
2answers
46 views

Is this a valid proof that $\{ax + by|x,y \in \Bbb Z\}= \{n\times \gcd(a, b) |n\in \Bbb Z\}$?

I'm trying to prove that $\{ax + by|x,y \in \Bbb Z\}= \{n\times \gcd(a, b) |n\in \Bbb Z\}$, but I'm unsure on the . The main proposition I'm using to solve this is that $\exists x,y, ax+by = \gcd(a, ...
3
votes
1answer
132 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
1
vote
2answers
196 views

TI83+. Math-> Solver. Why does it give different solutions?

Not sure if anyone still uses this one.... Recall the keystrokes: Math 0:Solver Up arrow Enter equation Press ENTER You will now be at a prompt displaying the equation, and a line saying x=### ...
4
votes
2answers
156 views

Solve $x^2 + 8 = 3^n$, where $x\in\mathbb N$.

First of all, $n$ is even (find that out by trying to see what kind of remainders you can possibly get when dividing everything by 4). So $n=2m$, $m\in N$. $ x^2 + 8 = 3^{2m} \Rightarrow x^2 - 1 = ...
1
vote
1answer
140 views

Heronian isosceles triangles

This is a problem from Project Euler, problem 94. The problem asks about isosceles triangles with integer sides (differing by 1 unit, e.g, 5-5-6) and integer area, which are known to be Heronian ...
4
votes
0answers
137 views

Find all the solutions of $x^2+7=2^n$. [duplicate]

Checking for some small natural numbers $n$, I found out that $2^n-7$ is a perfect square for $n=3,4,5,7,15$. How can we find all of the numbers $n$ for which $2^n-7$ is a perfect square? What I ...
0
votes
1answer
69 views

A question on integer values of an expression in positive rational variables

i) How do we find all $x,y ∈ \mathbb Q^+$ such that $x+y+ \dfrac1{x} +\dfrac1{y}$ is an integer ? ii)Let $p$ be an odd prime and $q=\dfrac{p-1}2$ , then how do we find all $x,y ∈ \mathbb Q^+$ such ...
17
votes
3answers
377 views

Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$ for positive integers $x$ and $y$?

Let $x$ and $y$ be positive integers. Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$? I ran a program for $1\le{x,y}\le1\text{ }000\text{ }000$ and found no solution, so I believe there are none.
1
vote
1answer
123 views

What kind of methods there are to solve a Diophantine equation from IMO longlist?

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get ...
0
votes
1answer
63 views

Reducing radical congruence to polynomial congruence

I am trying to find a way to describe all integer values of $x$ for which the following holds true: $\sqrt[2]{(1/2) * x * (x - 1) + (1/4)} + (1/2)\in \mathbb{Z}$ Noting that this can be equivalently ...