Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

189
votes
12answers
20k 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 ...
39
votes
5answers
1k views

Golden Number Theory

The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers ...
37
votes
5answers
698 views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
36
votes
3answers
1k 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? ...
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 ...
34
votes
6answers
1k 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 $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
34
votes
1answer
1k 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 ...
32
votes
2answers
496 views

Is $n(n+1)$ ever a factorial?

Brocard's problem asks if $(n-1)(n+1)$ is ever a factorial. My question is similar: is $n(n+1)$ ever a factorial? This can be seen as the special case $k=2$ of the question: for $2\le k\le n-2,$ when ...
31
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 $x^y = y^x$?
31
votes
2answers
865 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
773 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 ...
29
votes
2answers
628 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 ...
26
votes
5answers
608 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 ...
25
votes
2answers
479 views

Find $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares $ (s=abcde)$

Are there five distinct positive integers $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares ? $ (s=abcde)$ If ...
24
votes
2answers
2k 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 ...
23
votes
2answers
1k 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 ...
23
votes
4answers
655 views

Minimum of $n$? $123456789x^2 - 987654321y^2 =n$ ($x$,$y$ and $n$ are positive integers)

What is the minimum of $n$? $x$,$y$ and $n$ are positive integers, find the minimum of $n$, such that: $123456789x^2 - 987654321y^2 =n$
23
votes
5answers
437 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 ...
22
votes
1answer
1k 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 ...
21
votes
3answers
603 views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies ...
19
votes
8answers
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 ...
19
votes
2answers
319 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. ...
18
votes
6answers
2k 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 ...
18
votes
1answer
451 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 ...
17
votes
3answers
321 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.
17
votes
2answers
654 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
17
votes
3answers
891 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.
17
votes
1answer
458 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 ...
16
votes
2answers
613 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 ...
15
votes
3answers
1k 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 ...
14
votes
3answers
512 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
2answers
340 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 $\mathbb N$ ? $$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$
14
votes
2answers
182 views

Do there exist an infinite number of integer-solutions $(x,y,z)$ of $x^x\cdot y^y=z^z$ where $1\lt x\le y$?

Question : Do there exist an infinite number of integer-solutions $(x,y,z)$ of $x^x\cdot y^y=z^z$ where $1\lt x\le y$ ? Motivation : After struggling to find a solution, I've just got one solution, ...
13
votes
2answers
352 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
2answers
216 views

Solve $(a^2-1)(b^2-1)=\frac{1}4 ,a,b\in \mathbb Q$

Does the equation $(a^2-1)(b^2-1)=\dfrac{1}4$ have solutions $a,b\in \mathbb Q$? I search $0<p<1000,0<q<1000$, where $a=\dfrac{p}q$, but no solutions exist. I wonder is this equation ...
13
votes
1answer
429 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
4answers
400 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
13
votes
1answer
535 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 ...
13
votes
2answers
257 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
346 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? ( $\{x\}$ ...
13
votes
4answers
567 views

When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?

Is $$\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod{2^n} \ne 0\;$$ always true when $n \ge 3$. Baker's theorem on transcendental numbers that provide bounds for diophantine equations may be ...
12
votes
4answers
720 views

Proof that $x^2+4xy+y^2=1$ has infinitely many integer solutions

The question would, naturally, be very straight forward if there was a $2xy$ instead of a $4xy$. Then it would simply be a matter of doing: $$ x^2+2xy+y^2=1\\ (x+y)^2=1\\ \sqrt{(x+y)^2}=\sqrt{1}\\ ...
12
votes
4answers
400 views

Tricky positive diophantine equation

Find all positive integer solutions to the equation $x^3=y^5+100$. Notice that $7^3=3^5+100$ is a solution, but I can't find any more. Thanks for your help! P.S. I don't know any advanced number ...
12
votes
5answers
806 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
669 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 ...
12
votes
2answers
495 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 ...
12
votes
2answers
201 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
3answers
278 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 writing a computer program, I just tried ...
12
votes
1answer
292 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 ...
12
votes
1answer
201 views

A curious phenomenon for the cubic diophantine $x^3+y^3+z^3 = 1$?

Given positive integers $a,b,c$, $$a^3+b^3 = c^3\pm1\tag{1}$$ then there are exactly 133 solutions with $c<10^6$. (Wroblewski has tables for the general $x_1^3+x_2^3+x_3^3+x_4^3 = 0$ with all ...