Tagged Questions
5
votes
2answers
53 views
About the infinite solutions of a Diophantine equation
Consider the following problem:
$$\sum_{k=1}^N k^2=q^2$$
where q is an integer number. This can be written as:
$$\frac{1}{3}N^3+\frac{1}{2}N^2+\frac{1}{6}N=q^2$$
In the same way we can write:
...
1
vote
1answer
40 views
A Diophantine equation and decimal digits
Solutions of the Diophantine equation
$a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$
are
12=3*4,
56=7*8,
67100672=8191*8192.
Are there more solutions/examples like that or a generalization of the ...
9
votes
1answer
69 views
Examples of Diophantine equations with a large finite number of solutions
I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
2
votes
1answer
34 views
Solution of a Diophantine equation involving powers
Is it possible to show that, given $n$, there are infinite values of $k$ giving solutions of the equation:
$$x^n+ky^n=z^n$$
with $k,x,y,z,n$ natural numbers?
The constrains are: $$2\lt n, 1\lt k$$
7
votes
2answers
103 views
For which integers x, y is $2^x + 3^y$ a square of a rational number?
For which integers x, y is $2^x + 3^y$ a square of a rational number?
(Of course $(x,y)=(0,1),(3,0)$ work)
3
votes
1answer
109 views
Find all integer solutions to $x^2+4=y^3$.
Find all integer solutions to $x^2+4=y^3$.
Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
6
votes
2answers
134 views
Solve: $x^2-py^2=q$
Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers.
It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol.
...
4
votes
1answer
80 views
(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$
Find the integer solutions:
$$a·b^5+3=x^3,a^5·b+3=y^3$$
This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$.
Thanks in ...
0
votes
1answer
42 views
Linear Diophantine Equations: Integer Solutions $x,y$ exist for $ax+by=c$, but how do I find them by hand?
I'm trying to find which of $133x+203y=38$, $133x+203y=40$, $133x+203y=42$, and $133x+203y=44$ have integer solutions. I know that only the third equation suffices for these conditions because ...
27
votes
6answers
548 views
Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
5
votes
3answers
152 views
Integer solutions of $n^3 = p^2 - p - 1$
Find all integer solutions of the equation, $n^3 = p^2 - p - 1$, where p is prime.
2
votes
2answers
45 views
Sums that are pythagorean and normal
I noticed that
$3^2+4^2+15^2=9^2+13^2$
and also
$3+4+15=9+13$
Is there an easy way to find all pairs of disjoint sets of positive integers whose sum are the same and whose sum of squares are the ...
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 ...
0
votes
0answers
57 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 ...
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:
...
2
votes
1answer
101 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
132 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
63 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 ...
3
votes
4answers
137 views
Integral solutions of hyperboloid $x^2+y^2-z^2=1$
Are there integral solutions to the equation $x^2+y^2-z^2=1$?
7
votes
2answers
169 views
Solve : $\space 3^x + 5^y = 7^z + 11^w$
Solve the diophantine equation $3^x + 5^y = 7^z + 11^w$,here $x,y,z,w$ are all non-negative integers.
I find three solutions by force algorithm use Mathematica: (0,0,0,0)(1,1,1,0)(1,3,1,2).And ...
16
votes
3answers
320 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.
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 ...
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 ...
3
votes
1answer
90 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 ...
0
votes
5answers
165 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:
$$ ...
4
votes
1answer
74 views
Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$
Suppose you have an equation of the form
$$
a(n^2 - m^2) + b(n-m) + c = 0
$$
With given integers $a$, $b$ and $c$.
Is there a necessary and sufficient condition that the equation has no solutions ...
2
votes
2answers
187 views
$x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
How do I prove that the diophantine equation $x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
5
votes
2answers
115 views
Does there exist $a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
Does there exist $ a,b,c\in \mathbb Q$ such that $(a+b+c)^2 + 3(a+b+c)+5=2(ab+bc+ca)$
I think the answer is no
4
votes
1answer
206 views
$x^4 - y^4 = 2z^2$ has no solution
How do I prove that the equation $x^4 - y^4 = 2 z^2$ has no solutions using the fact that the equations $x^4 + y^4 = z^2$ and $x^4 - y^4 = z^2$ have no solutions.
I cant think of a method of reducing ...
2
votes
2answers
69 views
Twice a triangle is triangle
The question is to prove that there are infinitely many triangular numbers $T_n$ where $2 \times T_n$ is also a triangular number, and give the first few as an example.
My attempt:
$$2 \cdot {x(x+1) ...
3
votes
2answers
67 views
Factorials and Arithmetic Progression.
Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression.
$n$ is a natural number
I don't see any such examples(Except for $n=2$). And I don't see ...
16
votes
7answers
1k views
Pythagorean triplets $x^2+y^2 = z^3$
I need to prove that the equation $x^2 + y^2 = z^3$ has infinitely many solutions for positive $x, y$ and $z$.
I got to as far as $4^3 = 8^2$ but that seems to be of no help.
Can some one help me ...
4
votes
2answers
85 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 ...
15
votes
2answers
272 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 ...
10
votes
2answers
301 views
Solve $y^2= x^3 − 33$ in integers
This is not homework, could someone provide a nice clear proof as I have been struggling with this for some time.
Solve the equation $y^2= x^3 − 33$; $x, y \in \mathbb{Z}$
7
votes
2answers
90 views
Diophantine Quintuple?
I have come across the following set of numbers: $\{1, 3, 8, 120\}$
These are positive integers where the product of any two of the numbers equal to a number that is one less than a square number. ...
2
votes
2answers
83 views
Solve for system of diophantine equations
$\cases{x+1=a^2 \cr x^3-x^2+1=b^2}$
I just can found a trivial solution $x=0$. Is there any other ?
1
vote
2answers
124 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 ?
12
votes
2answers
176 views
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube.
I found $x=0$, any others ?
7
votes
4answers
176 views
Solve $a^3-5a+7=3^b$ over the positive integer
Solve $a^3-5a+7=3^b$ over the positive integer
I don't know how to solve such equation, please help me. Thanks
2
votes
4answers
127 views
is there any number pattern in the sum of square of two nos. and cube of 2 nos.
I wish to know the numbers which can be written in the form of sum of squares of two numbers and cube of two numbers and is there any pattern in it?
0
votes
2answers
103 views
Non Linear Diophantine Equation in Three Variables
Find all positive integer solution to $abc-2=a+b+c$.
3
votes
1answer
60 views
Find $ k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer root
Find $k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer roots
I know a similar problem $x^3 + y^3 + z^3 = nxyz$
but I still can't solve my problem
5
votes
1answer
162 views
$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)
$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$
$$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
2
votes
1answer
121 views
Diophantine equations - Perfect square and Perfect cube related
Solve following Diophantine equations:
$1) \ a^3-a^2+8=b^2$
2) $a, \ b,\ c \in \mathbb{Z^+}$$$\frac{a^3}{(b+3)(c+3)} + \frac{b^3}{(c+3)(a+3)} + \frac{c^3}{(a+3)(b+3)} = 7$$
3) $a^3-8=b^2$
In ...
1
vote
1answer
78 views
Like Diophantine equation
The equation $x^n - ny^x-nxy$ = $0$ has solution set $(n, x, y) = (1, 1, \frac12), (2, 1, \frac14), (3, 1, \frac16), \ldots$
I would like to know/learn the following (Kindly discuss)
1) If we ...
4
votes
3answers
147 views
Find $a, b, c, d \in \mathbb{Z}$ such that $2^a=3^b5^c+7^d$
Solve $2^a=3^b5^c+7^d$ over the positive integer.
I know $a$ is even because: $(-1)^a \equiv2^a = 3^b5^c+7^d \equiv1 \ (mod\ 3)$
0
votes
0answers
42 views
Diophantine, elliptic analysis
We have seen the Erdös-Straus conjecture relating to the theory of elliptic curves.
How to study and analyze asymptotic estimates on Diophantine equation?.
How to use the theorem of ...
2
votes
2answers
88 views
How can I find the integer solutions to $x^2+x-2y^2=0$?
I enter this equation in Wolfram Alpha : $x^2+x-2y^2=0$
and it gave me something like this :
and I am wondering how this solution is found and how to know if a given equation would guarantee to ...
