Questions on finding integer/rational solutions of equations.
151
votes
13answers
17k views
Unusual 5th grade problem, how to solve it
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was ...
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 ...
36
votes
3answers
942 views
Triangular Factorials
I came across a statement online and have been looking for a proof :
It states that 1, 6 and 120 are the only numbers which are both triangular and factorials.
Is there any way I can prove this? ...
32
votes
1answer
993 views
Decomposing polynomials with integer coefficients
Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
29
votes
2answers
698 views
Any positive integer solutions to $x^6+y^{10}=z^{15}$?
This question might be easy.
The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, ...
29
votes
2answers
653 views
Does $2x^2-1=y^5$ have a solution in integers, with $y>1$?
In my solution to this MSE problem, I noted that $2x^2-1=y^5$ is unlikely to have
solutions in integers with $y>1$. Recently, I've tried to find a proof, without success.
Following Thomas ...
28
votes
5answers
2k views
$x^y = y^x$ for integers $x$ and $y$
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
27
votes
6answers
567 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
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
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 ...
23
votes
5answers
551 views
When is $1^5 + 2^5 + \ldots + n^5$ a square?
When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$.
I feel that the ...
22
votes
2answers
964 views
Diophantine applications of Spec?
Let $f(\bar x)$ be a multivariable polynomial with integer coefficients.
The zeros of that polynomial are in bijection with the homomorphisms $\mathbb Z[\bar x] \rightarrow \mathbb Z$ that factor ...
21
votes
2answers
1k views
Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?
The background of this question is this: Fermat proved that the equation,
$$x^4+y^4 = z^2$$
has no solution in the positive integers. If we consider the near-miss,
$$x^4+y^4-1 = z^2$$
then this has ...
18
votes
6answers
901 views
Integral solution for $|x | + | y | + | z | = 10$
How can I find the number of integral solution to the
equation
$|x | + | y | + | z | = 10.$
I am using the formula,
Number of integral solutions for $|x| +|y| +|z| = p$ is $(4P^2) +2 $, So the ...
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 ...
16
votes
1answer
324 views
Integer solutions of $x! = y! + z!$
There was an interesting problem asked about triples $(x,y,z)$ which are solutions of
$$x! = y! + z!.$$
Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$.
Now I ...
16
votes
2answers
279 views
All positive integer solutions to $\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$
As the title states, how would I go about finding the positive integer solutions of
$$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$$?
Thank you for your help.
...
16
votes
3answers
322 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.
15
votes
3answers
960 views
Are there integer solutions to $9^x - 8^y = 1$?
This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.
Edit:
Of course, $x = 1, y = 1$ is a ...
15
votes
1answer
394 views
Finding solutions to equation of the form $1+x+x^{2} + \cdots + x^{m} = y^{n}$
Exercise $12$ in Section $1.6$ of Nathanson's : Methods in Number Theory book has the following question.
When is the sum of a geometric progression equal to a power? Equivalently, what are ...
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 ...
14
votes
3answers
462 views
$x,y$ are integers satisfying $2x^2-1=y^{15}$, show that $5 \mid x$
Let $x, y >1$ be integers satisfying $2x^2-1=y^{15}$. How can I prove that $5 \mid x$?
14
votes
1answer
447 views
How to compute rational or integer points on elliptic curves
This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
14
votes
2answers
333 views
Solutions of $q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$ s.t. $q \geq 3$
Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ?
$$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
13
votes
2answers
334 views
Solution in integers to $2^n+n=3^m$
How to find all positive integers $m,n$ such that $2^n+n=3^m$ ?
We have by inspection $(m,n)=(0,0)$ and $(1,1)$
And there are no more for m and n both less then $100$.
13
votes
1answer
417 views
Positive integers satisfying $x^{a+b} = a^b \cdot b$, how to show that $a=x$ and $b=x^x$?
Let $a,x,b$ be positive integers satisfying $x^{a+b} = a^b \cdot b$. How can I prove that $a=x$ and $b=x^x$?
13
votes
2answers
230 views
How did Letac solve $x_1^k + x_2^k + \dots +x_9^k = 0$ for $k = 1, 3, 5, 7$ in 1942?
It's quite easy to find integer solutions to,
$$x_0^k + x_1^k + \dots +x_9^k = 0$$
for $k = 1, 3, 5, 7$. One I found is, if $x^2-10y^2 = 9$, then,
$$1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + ...
13
votes
1answer
288 views
Roots with equal fractional parts
Question. ¿Does there exist an integer $n>1$ such that there exist positive integers $a,b$ such that $\{\sqrt[n]{a}\}=\{\sqrt[n]{b}\},a\neq b$ and $a$ and $b$ aren't perfect n-th powers? ( $\{x\}$ ...
12
votes
5answers
728 views
Another quadratic Diophantine equation: How do I proceed?
How would I find all the fundamental solutions of the Pell-like equation
$x^2-10y^2=9$
I've swapped out the original problem from this question for a couple reasons. I already know the solution to ...
12
votes
2answers
176 views
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube
Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube.
I found $x=0$, any others ?
12
votes
1answer
344 views
Find the positive integer solutions of $m!=n(n+1)$
Find the positive integer solutions of $m!=n(n+1)$
I basically have $(m,n)=(2,1)$ or $(3,2)$ and I think these are the only solutions.
I don't have a complete proof but here's what I know so far.
By ...
12
votes
1answer
274 views
$a^m+k=b^n$ Finite or infinite solutions?
Given positive integers k,a,b, is there a finite or infinite number of solutions in positive integers $m,n>1$, to $a^m+k=b^n$?
Pillai's conjecture states that each positive integer occurs only ...
11
votes
2answers
504 views
What is the simplest ellipse that goes through exactly 13 lattice points?
The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points.
Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
11
votes
3answers
310 views
Non-squarefree version of Pell's equation?
Suppose I wanted to solve (for x and y) an equation of the form $x^2-dp^2y^2=1$ where d is squarefree and p is a prime. Of course I could simply generate the solutions to the Pell equation ...
11
votes
2answers
393 views
Integral solutions of $x^2+y^2+1=z^2$
I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
11
votes
2answers
206 views
How to find a “better description” (e.g. recurrence relation) for this sequence?
My solution to a problem in Project Euler required to solve this subproblem: find values of $k\in\mathrm{N}$ such that $3k^2+4$ is a perfect square.
As I was writting a computer program, I just tried ...
10
votes
2answers
382 views
Machin's formula and cousins
There exists a well-known formula by John Machin:
$$\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}.$$ Actually, it belongs to the family of Machin-like formulas of the form
...
10
votes
2answers
303 views
Solve $y^2= x^3 − 33$ in integers
This is not homework, could someone provide a nice clear proof as I have been struggling with this for some time.
Solve the equation $y^2= x^3 − 33$; $x, y \in \mathbb{Z}$
10
votes
5answers
423 views
How to find solutions of $x^2-3y^2=-2$?
According to MathWorld,
Pentagonal Triangular Number: A number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$. Such numbers exist when
...
10
votes
3answers
171 views
Triples of Numbers
I have a question:
How many triples $(a,b,c)$ are there such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$$ and $a <b<c$? They have to be positive integers. Also find those triples.
I know ...
10
votes
3answers
138 views
For which n are there primitive Pythagorean triples with legs of lengths a and a+n?
For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there ...
10
votes
2answers
315 views
diophantine equation in positive integers
would you please help me solve this?
solve this equation in positive integers:
$x^2+y^2+z^2=3xyz$
I could prove that it's solutions are infinite, for if $(x,y,z)$ is a solution, with $x\le y \le ...
10
votes
1answer
173 views
Diophantine equation $x^y-y^x=11$
How can one find all integer solutions to $x^y-y^x=k$, for a given k?
Example case $x^y-y^x=11$
9
votes
2answers
1k views
Three variable, third degree Diophantine equation
I haven't found any useful method to solve the following problem: Prove that if $x,y,z\in\mathbb{Z}$ and $x^3+y^3=3z^3$ then $xyz=0$.
Source: ...
9
votes
2answers
166 views
Determining the number $N$
Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
9
votes
1answer
153 views
How to find all integers $a,b > 1$ satisfying $b \mid a^2+1$ and $a^2 \mid b^3+1$?
Let $a,b\in \mathbb{Z}$ with $a,b>1$, and such that $b \mid a^2+1$ and $a^2 \mid b^3+1$. Find all such $a,b$.
I found $a=3,b=2$. Are there any other solutions? Thank you.
yesterday I have ...
9
votes
1answer
74 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 ...
9
votes
1answer
136 views
Local solutions of a Diophantine equation
I am trying to prove that the equation
$$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$
has a non-trivial solution for all primes $p$.
I am sure that this is a standard exercise, and I have done the easy ...
9
votes
1answer
389 views
Exponential Diophantine Equations for Beginners
What would be some exponential Diophantine equations for the beginner to solve (which can demonstrate the techniques?) especially good if there are hints! Thank you very much!
8
votes
5answers
397 views
Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$?
Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$?
I'm fairly certain it does, but I'm hoping to see a rigorous proof of this statement. Thanks.
Here is my motivation. I'm ...
8
votes
1answer
331 views
Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$?
$x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
