Questions on finding integer/rational solutions of equations.
2
votes
2answers
171 views
rational triangles and cosines
I've recently started to try working on exercises from this book on Diophantine equations before I need to return it to the library. This one has me slightly stumped. It asks to show that the cosine ...
0
votes
0answers
43 views
Wrong answer on elementary diophantine equation - why?
Solve the equation and show all possible, non-negative values for X
and Y: $5X+4Y=60$
So I wanted to do it like that:
$$5X+4Y=60\leftrightarrow0X+4Y=0 \pmod5$$, thus $4Y=5k$ where $k\in Z$. ...
2
votes
1answer
43 views
Number of integer solutions of $xy - 6 (x+y)=0$
What are the number of integer solutions of $xy - 6 (x+y)=0$ with $x\leq y$ is ?
Equation $xy - 6 (x+y)=0$ can also be written as $1/x + 1/y = 1/6$
1
vote
0answers
73 views
Solve the equation $x^4+y^4=d*z^2$
Solve the equation:$$x^4+y^4=d*z^2,$$
where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer.
I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
1
vote
0answers
69 views
On the elliptic curve $x^4+y^4 =193z^2$
Given the simultaneous Diophantine equations,
$$u^2+v^2=w^2\tag{1}$$
$$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$
the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the ...
2
votes
2answers
77 views
No rational solutions of a system of equations
Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that
\begin{matrix}
a^2b+2b^2c+2ac^2=0\\
a^2c+ab^2+2bc^2=0\\
a^3+2b^3+4c^3+12abc=3.
\end{matrix}
I'm able to show that this ...
36
votes
5answers
1k views
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$
In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of ...
7
votes
3answers
633 views
Proving a statement regarding a Diophantine equation
FINAL EDIT : Prove that if $p^z|n^2-1$
$$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$.
Here $p>3$ is an odd prime , $x=2y+z, \ ...
0
votes
3answers
46 views
Is there a squared matrix $A$ sized 2x2 that follows the next criteria?
Is there a squared matrix $A$ sized 2x2 that it's elements $\in \mathbb Z$ so that
$$
A^2 =
\begin{pmatrix}
2 & 3 \\
2 & 4 \\
\end{pmatrix}
$$
0
votes
0answers
59 views
How to find integer solutions of an equation using approximation methods?
If I have a function called $f(x)$ that have several roots, integers and not integers. How can I find just the integer ones by approximations methods?
A simple example would be ...
3
votes
2answers
63 views
Elementary Diophantine equation
Solve $(x+y)(xy+1)=2^z$ in positive integres. My attempts is to use $x+y=2^a$, $xy=2^b-1$ and therefore $x,y$ are the roots of the quadratic equation $w^2-2^aw+2^b-1=0$. I try to analyze its ...
1
vote
1answer
80 views
Suppose $x$, $y$, and $z$ are integers that satisfy the system equations
Suppose $x$, $y$, and $z$ are integers that satisfy the system equations :
$x^2y$+ $y^2z$ + $z^2x$ $= 2186$
$y^2x$+ $z^2y$ + $x^2z$ $= 2188$
What is $x^2+y^2+z^2$ ?
3
votes
4answers
60 views
Solve : $\frac{n}{2}(n+1)=2014+2k$.
$n,k$ are positive integers and $n>k$, solve the equation : $$\frac{n(n+1)}{2}=2014+2k.$$
the first thing I did is to write the LHS as $(2n+1)^2$ but I face an equation like $ak+b=m^2$, I know ...
5
votes
2answers
175 views
Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$
Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$.
The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and ...
1
vote
2answers
94 views
How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory
I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units ...
8
votes
2answers
93 views
Solve : $ab(a+b)(a-b)=c^2-1$
As we know that $ab(a+b)(a-b)=c^2$ has no integer solution in $Z^+$.However, it seems that $$ab(a+b)(a-b)=c^2-1$$
has infinite positive integer solutions,could you prove it?
Here are some of them:
...
5
votes
3answers
718 views
Integral solutions to $y^{2}=x^{3}-1$
How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$.
I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot ...
2
votes
1answer
106 views
If $x^p +y^p = z^p$ and $xyz \neq 0$, then $p$ divides $x$ or $y$ or $z$?
I am working on an exercise: If $x^5 +y^5 = z^5$ and $xyz \neq 0$, then $5$ divides at least one of $x$, $y$ or $z$.
I am thinking that the answer involves an application of Kummer's theorem, but I'm ...
4
votes
1answer
133 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 ...
2
votes
2answers
64 views
Show that the equation $x^{13} +12x + 13y^6 = 1$ doesn't have integer solutions
So I'm asked to show that the equation $x^{13} + 12x + 13y^6 = 1$ doesn't have integer solutions.
I'm not quite sure how to approach the problem as this doesn't seem to look like anything I had in ...
12
votes
1answer
344 views
Find the positive integer solutions of $m!=n(n+1)$
Find the positive integer solutions of $m!=n(n+1)$
I basically have $(m,n)=(2,1)$ or $(3,2)$ and I think these are the only solutions.
I don't have a complete proof but here's what I know so far.
By ...
3
votes
4answers
143 views
Integral solutions of hyperboloid $x^2+y^2-z^2=1$
Are there integral solutions to the equation $x^2+y^2-z^2=1$?
3
votes
1answer
91 views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
24
votes
1answer
407 views
Finding integer solutions for trigonometric equation $8\sin^2\left(\frac{(k+1)\pi}{n}\right)=n\sin\left(\frac{2\pi}{n}\right)$
I thought up the problem of finding a regular $n$-sided polygon that has a diagonal with lenght $d_k$ such that the area of the polygon equals ${d_k}^2$. By doing some easy trigonometry within the ...
8
votes
4answers
211 views
Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$
Let's say you're asked to find a Pythagorean triple $a^2 + b^2 = c^2$ such that $a + b + c = 40$. The catch is that the question is asked at a job interview, and you weren't expecting questions about ...
16
votes
3answers
323 views
Finding all integer solutions of $5^x+7^y=2^z$
Find all integers $x,y,z$ such that $5^x+7^y=2^z$.
This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists.
1
vote
1answer
96 views
Alternative solutions to $n^2+n = k^2+k + 2kn$
Consider this equation:
$n^2+n = k^2+k + 2kn$
I want to find the set of non-negative integer n,k that satisfies the equation.
I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
0
votes
0answers
11 views
Which diophantine polynomials generate these diophantine sets?
Via Matiyasevich's Theorem, it is easy to prove that the following sets are diophantine:
$\{k\}$
$\{0, 1, \dots, k-1, k+1, k+2, \dots \}$
$\{0, 1, \dots, k\}$
$\{k+1, k+2, \dots\}$
Number 1 is ...
6
votes
3answers
212 views
Diophantine equation $x^2 + 32x = y^3$
I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$.
I think that the first step is to factorise: $x(x+32)=y^3$.
If $x$ is odd, then $x+32$ is also odd. The common ...
0
votes
5answers
166 views
Generate solutions of Quadratic Diophantine Equation
Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation
Here's the answer:
$$ ...
3
votes
2answers
77 views
Is it possible to solve for two unknowns from one equation?
Is it possible to solve for two unknowns using only one equation?
For example:
$x+3y=32$
Where $x$ and $y$ are integers.
Thanks :)
7
votes
3answers
200 views
Proving there are no integers $a, b, c$ satisfying $12a + 18b + 27c = 227$
Given $12a + 18b + 27c = 227$, how can we prove that $a, b, c$ can never be integers? I don't have many ideas. Can someone give me some ideas?
6
votes
4answers
627 views
Proving that an integer is the $n$ th power
I have not been able to solve this problem. Any insights would be appreciated!
Let $x, n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_{k}$
such that $x − a_k^n$ is ...
0
votes
3answers
53 views
Solving the algebraic equation
I am trying to solve this: $$x-40={-400\over x}$$
The answer must be $x=20$
Please give step by step explanation.
1
vote
2answers
298 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
2answers
58 views
How to solve $5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$ in $\mathbb{Z}$
how to solve in $\Bbb Z$:
$$5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$$
3
votes
3answers
173 views
How to solve Quadratic Diophantine Equation
Here's the problem.
Find the solutions of the following equation:
$$ k^2 - 1 = 5(m^2 - 1).$$
Here's my idea:
The original equation can be written as:
$$ k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ...
2
votes
2answers
193 views
$x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
How do I prove that the diophantine equation $x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
4
votes
1answer
77 views
Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$
Suppose you have an equation of the form
$$
a(n^2 - m^2) + b(n-m) + c = 0
$$
With given integers $a$, $b$ and $c$.
Is there a necessary and sufficient condition that the equation has no solutions ...
1
vote
1answer
126 views
$x^4+y^4=2z^2$ has only solution, $x=y=z=1$ .
How do I verify that the only solution in relatively prime positive integers of the equation $x^4+y^4=2z^2$ is $x=y=z=1$?
0
votes
0answers
28 views
Non-negative integral solutions to a single equation
Assume we have a integral vector $c\in \mathbb{Z}^n$ and an integer constant $b\in \mathbb{Z}$. Is there a necessary and sufficient condition for whether or not there exists a non-negative integer ...
5
votes
2answers
116 views
Does there exist $a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
Does there exist $ a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
I think the answer is no
4
votes
1answer
210 views
$x^4 - y^4 = 2z^2$ has no solution
How do I prove that the equation $x^4 - y^4 = 2 z^2$ has no solutions using the fact that the equations $x^4 + y^4 = z^2$ and $x^4 - y^4 = z^2$ have no solutions.
I cant think of a method of reducing ...
2
votes
2answers
69 views
Twice a triangle is triangle
The question is to prove that there are infinitely many triangular numbers $T_n$ where $2 \times T_n$ is also a triangular number, and give the first few as an example.
My attempt:
$$2 \cdot {x(x+1) ...
0
votes
0answers
25 views
Sphere containment problem inside a rational convex polytope of general dimensions.
Given a positive number $r$ and a rational convex polytope (bounded polyhedra) described by its set of half-planes (system of linear inequalities: $A\cdot x \leq b$, where $A\in\mathbb{R}^{m\times ...
1
vote
2answers
126 views
Solve for diophantine equation $x^n + y^n + z^n =1$ [closed]
Solve for diophantine equation
$x^n + y^n + z^n =1$
$x^n+y^n+z^n=2$
Is this equation solve-able ?
15
votes
2answers
274 views
$(x-a)(x-b)(x-c)(x-d)=ex$
We can verify that $x=125,162,343$ are the roots of equation $(x-105)(x-210)(x-315)=2584x$.
My question is,Could you find five positive integers $a,b,c,d,e$, which $(x-a)(x-b)(x-c)(x-d)=ex$ has four ...
4
votes
2answers
86 views
The rational points on the curve: $y^2=ax^4+bx^2+c$.
I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$.
Is there infinite rational points on this curve?
For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, ...
1
vote
0answers
87 views
Week of the problem on Diophantine equation
S.E board!
This is a Diophantine equations problem, which is so interesting one can do by plugging the suitable values in unknown. When it comes for finding set of all solutions is may be tough. I ...
0
votes
0answers
30 views
How do I find the set of integers solving a system of equations that contain outliers?
I have a system of $s$ equations that should (but won't) all equal some real unknown scalar value, $x$:
$x = v_1*k_1 + a_1*k_1*m = v_2*k_2 + a_2*k_2*m = ... = v_s*k_s + a_s*k_s*m$
where,
$k_i$ are ...
