Questions on finding integer/rational solutions of polynomial equations.

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57 views

Woking Heron's Formula In Reverse

I'm writing a program to generate randomized Heron's Formula word problems. I need to figure out how to work the problem in reverse so that the answer will come out to an integer. As an example, if I ...
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1answer
36 views

Buying dogs and Cats and mice

This problem is only considering positive integer solutions: You must spend exactly 100 and purchase exactly 100 animals. Each dog costs 15 and each cat costs 1 and each mouse costs .25. How many of ...
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2answers
931 views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
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1answer
19 views

FLT and non-maximal orders

The ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is the unique maximal order of that field. Kummer's attempt at proving FLT fails for prime exponents which are irregular, i.e. divide ...
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2answers
32 views

Diophantine Equation Related To Triangles

a,b and c are the sides of a triangle and a, b, c are integers. I need to solve the following Diophantine equation for positive integral values of k. $bc(b+c-a) = k^{2}(a+b+c)$ I think some ...
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2answers
32 views

Find integer solutions of $(1) xy=2x+2y$ and $ (2)xy=2x+y.$

I've tried this for the first one:$xy=2(x+y).$ Therefore either x or y is divisible by 2. And I'm totally stuck on the second. How to solve these?
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2answers
260 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
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2answers
46 views

Number of integer solutions of two similar equations

Find the number of integer solutions of: (a) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{20}}$$ (b) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{2014}}$$ I know the ...
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0answers
50 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
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3answers
48 views

Show that there are only trivial solutions

How can I show that the only solutions of the diophantine equation $x^2+y^2=1$ are the trivial ones: $(x,y)=(0,1), (0,-1), (1,0), (-1,0)$ ? That's what I thought: $$x \equiv 0,1 \pmod 2 \Rightarrow ...
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1answer
26 views

Is there also an other way, to prove that the diophantine equation has no solution?

I am looking at the following exercise: The diophantine equation $y^2=x^3+7$ has no solution. If the equation would have a solution, let $(x_0,y_0)$,then $x_0$ is odd. ( If $x_0$ is even, $x_0=2k ...
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0answers
23 views

Diophantine equation with three variables

I'm trying to solve a diophantine equation with $3$ variables. The problem can be written as a system of equations: $\begin{cases} 0.5x + 4y + 9z &= 97 \\ x+y+z &= 34 \end{cases} \implies ...
2
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2answers
103 views

What is the density of solutions for Pythagoras' and Fermat's equation $x^2+y^2=z^2$

It is now proved that, for integer $n\geq 2$, the equation $x^n+y^n=z^n$ has integer solution only when $n=2$. When $n=2$, this equation has an infinity of solutions. My question is whether there is ...
2
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1answer
97 views

Homogeneous diophantine equation

Here I have a diophantine equation featuring a homogeneous polynomial: $$x^2+5y^2+34z^2+2xy-10xz-22yz=0; x, y, z\in\mathbb{Z}$$ I have no idea how to approach this, I've tried various substitutions ...
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2answers
191 views

3 Variable Diophantine Equation

Find all integer solutions to $$x^4 + y^4 + z^3 = 5$$ I don't know how to proceed, since it has a p-adic and real solution for all $p$. I think that the only one is (2, 2, -3) and the trivial ones ...
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0answers
70 views

Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ ax^2+bxy+cy^2+dx+ey+f=0 $$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
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3answers
136 views

Non-linear diophantine equation

Let $k$ and $n$ be positive integers and $y(n-x)=(k+nx)$. What is the condition of $k$ and $n$ such that there exist positive integers $x, y$ as the solution of $y(n-x)=(k+nx)$?
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1answer
36 views

$f(x)=x^3+ax^2+bx+p=0$ has no integral solutions.

let $p$ be a prime number, does the polynomial:$f(x)=x^3+ax^2+bx+p=0$ have any integral solution if $p>a>2$ and $ x>2 $? I concluded that there was none on the basis that $p>a$ and a ...
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1answer
89 views

$x^4-2x^3+x=y^4+3y^2+y$ in the set of integers

The task is to solve the equation $x^4-2x^3+x=y^4+3y^2+y$ in integers. I expect is has something to do with factorizing but have no concrete idea; any help? thx guys
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1answer
348 views

Is there any integer solutions of $3x^3+3x+7=y^3$?

$3x^3+3x+7=y^3$ $x, y \in \mathbb{N}$ Having thought about it two hours, and I'm still not sure how to show there actually aren't any integer solutions. EDIT Another formulation of this problem: ...
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1answer
57 views

Simpler Way to Solve This Diophantine Equation

In a pet shop, rats cost $5$ dollars each, guppies cost $3$ dollars each, and crickets cost $10 $ cents each. $100$ animals are sold, and the total cost is $100$. How many rats, guppies, and crickets ...
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1answer
38 views

Proof that $(ab+cd)^{\frac{1}{n}}$ is irrational?

Let $a,b,c,d,n >2, \gcd(a,b,c,d)=1$, how can I prove that $\sqrt[n]{ab+cd}$ is irrational if $\sqrt[n]{a},\sqrt[n]{b},\sqrt[n]{c},\sqrt[n]{d}$ are irrational? Any hint?
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2answers
61 views

How do you solve $k(a^2-b^2)=2(ax-by)$?

let $a,b,c,d,x,y,k$ be all non-zero positive integers >1. If $a^2-b^2 \neq0$,how do you find all the pairs $(x,y)$ such that $k(a^2-b^2)=2(ax-by)$. I have found so far only solutions where ...
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3answers
51 views

Integer solutions to $k^2m^2 -k^2 - m^2 +1 = n^2$

Can the positive integer solutions to $$ k^2m^2 -k^2 - m^2 +1 = n^2 $$ be characterized (in the sense that the solutions to $a^2+b^2 = c^2$ are characterized by $a=r^2-s^2, b=2rs, c=r^2+s^2$ with ...
2
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0answers
81 views

Mordell Diophantine: $x^2+11=y^3$

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
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2answers
174 views

When $x^2+6xy+y^2$ a square number?

Find all natural numbers $x$ and $y$ such that $x^2+6xy+y^2$ is a square number. For example, $(x,y)=(2,3)$ or $(x,y)=(3,10)$. Obviously, we can consider $gcd(x,y)=1$.
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4answers
295 views

Solving Diophantine equations involving $x, y, x^2, y^2$

My father-in-law, who is 90 years old and emigrated from Russia, likes to challenge me with logic and math puzzles. He gave me this one: Find integers $x$ and $y$ that satisfy both $(1)$ and $(2)$ ...
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3answers
180 views

A conjecture on products/composition of Pell forms

Based on a few brute-force calculations, I've formulated the following. Conjecture. Let $x,y,u,v,p,q,a,b,c \ge 2$ be integers such that $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, $$ and write \begin{align} ...
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1answer
72 views

Pell's equation for n=2

If know that $x=3$, $y=2$ is a solution of $$x^2-2y^2=1,$$ then apparently all other solutions can be calculated as $$x_k+y_k\sqrt{2}=(x+y\sqrt{2})^k,$$ which I have trouble understanding. I've been ...
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1answer
51 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
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1answer
33 views

if $k>1$, Does $a+b =k(ax+by)$ have finitely many solutions?

Let $a,b,k,x,y$ be non-zero integers, solve $a+b=k(ax+by)$. It's a rather simple problem, but I just want to make sure that I have got all the possible solutions.
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1answer
49 views

Solving for a variable in an integer divisibility problem

Say I have a problem of the form Where , , and are known integers, is some unknown variable, and is an integer output. Is there an approach I could take to determine if there is some integer ...
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2answers
73 views

Solve the equation $a+b+c=abc$ for $a,b,c\in\mathbb{Z}$

Solve for $a,b,c$ (where $a$, $b$, and $c$ are integers) the equation $$a+b+c=abc.$$ I would prefer a solution using trigonometry and I think that it might use the formula $\tan A + \tan B + \tan ...
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1answer
65 views

Solving algebraic equations for x

So I was able to find the least common denominator which is $12$ but I'm struggling to solving the equation: $$\frac{4(x - 2)}{6} - \frac{2(x + 4)}{4} = -\frac{2}{3}.$$
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107 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
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3answers
113 views

Solving $y^3=x^3+8x^2-6x+8$

Solve for the equation $y^3=x^3+8x^2-6x+8$ for positive integers x and y. My attempt- $$y^3=x^3+8x^2-6x+8$$ $$\implies y^3-x^3=8x^2-6x+8$$ $$\implies ...
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0answers
31 views

Characterize the set that solves the Diophantine inequality

Given that $a, b, c$ are integers and $2\max(|b|, |3a + b|) \le \min(|c|, |3a+2b+c|)$ What characterizes the solution set in $\mathbb{Z}^3$? Obviously this is equivalent to the system of linear ...
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64 views

To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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2answers
86 views

Parametric solution of the Diophantine equation $\frac{1}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} ,x,y,z∈Z^+.$

I have prove that, for any given positive integer $p,$ parametric solution of the Diophantine equation $$\frac{1}{p}=\frac{1}{x}+\frac{1}{y}$$ can be written in the form $x=ac(a+b),y=bc(a+b),$ where ...
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0answers
54 views

Integral point on certain cubic surfaces and rational parametric solutions

The motivation of the following question comes from the Problem D4 in the book "unsolved problems in number theory" by Richard Guy. No integral points on the surface $S:x^3+y^3+z^3=3$ is known other ...
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1answer
69 views

Diophantine equation: $2(x^3+xy+y^3)=3(x+y)$

Here is a nice equation: $2(x^3+xy+y^3)=3(x+y)$ over $ \mathbb{Z}$ x $\mathbb{Z}$. Any nice way to approach this?
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6answers
100 views

The diophantine equation $a^2+ab-b^2=0$

I first tried with brute force with $-1000 \leq a,b \leq 1000$ but found no solution. But then a simple argument showed me that there was no solution. Not only in the integers, but even for the ...
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2answers
100 views

Solve $a^2+b^n=c^2$

Let $a,b,c$ be co-prime integers >1, for all $n>2$, I need help finding the integral solutions of the diophantine equation $a^2+b^n=c^2$. I saw the result but I am curious about to how to get ...
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1answer
208 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how ...
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4answers
63 views

Finding the number of two digit numbers

I was solving questions from a book and it had a question : Find all two digit numbers such that the sum of digits constituting the number is not less than 7; the sum of squares of digits is not ...
5
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3answers
326 views

Generating Functions and Linear Diophantine Inequalities

The following exercise is from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick, page 46. A $k$-composition of $n$ is an ordered $k$-tuple of non-negative integers whose sum is $n$. ...
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1answer
195 views

Minimum of $|az_x-bz_y|$

I am trying to minimize the following function: \begin{align} &f(z_x,z_y)=|az_x-bz_y| \\ &\text{ s.t. } z_x,z_y \in \mathbb{Z},1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \text{ and } ...
0
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4answers
71 views

Solve $c^2-b^2-a^2=2N$

Is there anyone that can help solving this equation: $c^2-b^2-a^2=2N$ where $a,b,c,N$ are natural numbers. Edit: We need to express $a,b,c$ for a certain $N$. Regards
0
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4answers
58 views

An example of how to solve equation over $\mathbb{Z}$

I found an example of how to solve equation over $\mathbb{Z}$ Example Solve the equation over $\mathbb{Z}$ : $$xy + 1 = 3x + y. $$ $$ xy = 1 + 3x + y \Longleftrightarrow (x-1) (y-3) = ...
5
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0answers
127 views

Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as ...