Questions on finding integer/rational solutions of equations.

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259
votes
14answers
31k views

Find five positive integers whose reciprocals sum to $1$

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 ...
56
votes
6answers
6k 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 $x^y = y^x$?
51
votes
4answers
1k views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides $(...
49
votes
5answers
2k 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 $\mathbb{Z}[\varphi]$...
48
votes
4answers
43k views

How to find solutions of linear Diophantine ax + by = c?

I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ? I tried to play around with it: $...
44
votes
2answers
2k 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 ...
42
votes
6answers
2k 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 $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
42
votes
5answers
1k 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 k\Big)^2-3\Big(\...
39
votes
3answers
2k 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 ...
37
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? ...
37
votes
2answers
646 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 ...
35
votes
2answers
1k 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, ...
33
votes
4answers
1k views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below $...
32
votes
0answers
561 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = \Big(1+\frac{...
31
votes
3answers
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 ...
30
votes
2answers
796 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 ...
30
votes
0answers
782 views

How to solve this two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I can not progress anymore......
29
votes
2answers
1k 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 Andrews'...
28
votes
2answers
539 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 $a=1,b=2,c=12,d=2380,s=...
27
votes
2answers
2k 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 ...
27
votes
1answer
3k 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 ...
27
votes
1answer
1k 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 ...
27
votes
1answer
558 views

For integers $a\ge b\ge 2$, is $f(a,b) = a^b + b^a$ injective?

Given two integers $a \ge b \ge 2$, can we encode them as a unique integer $a^b + b^a$? This question was asked a few weeks ago, but did not rule out the trivial cases. For example, if we allow ...
26
votes
5answers
693 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 identity$$\...
26
votes
3answers
3k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
25
votes
4answers
770 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: $123456789x^2 - 987654321y^2 =n$
25
votes
2answers
390 views

A diophantine equation with only “titanic” solutions

I made a note some time ago that I had read in some book that the equation $$313(x^3+y^3)=t^3$$ has positive integer solutions, but that these are so large that it would be absolutely hopeless to ...
25
votes
6answers
616 views

Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?

This following Problem is from Pell equation chapters exercise Let $y>3$ positive integer numbers, show that following diophantine equation $$x^2+x+1=7^y\tag{1}$$ has no integer solutions. ...
24
votes
8answers
856 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 ...
23
votes
3answers
2k views

Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align} &2 \times 2 = 2 + 2\\ ...
23
votes
3answers
763 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 ...
22
votes
8answers
2k views

Variation of 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 ...
22
votes
8answers
2k views

Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
21
votes
6answers
4k 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 ...
21
votes
6answers
4k views

Chicken Problem from Terry Tao's blog (system of Diophantine equations)

This problem was posted by Terry Tao in his blog earlier. It's actually from his son's Math Circle. It took him $15$ minutes to solve it. I guess we all can take a crack at it. Three farmers were ...
21
votes
1answer
573 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 ...
21
votes
5answers
6k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
21
votes
4answers
4k views

The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

Can someone provide the proof of 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 briefly as possible? I have seen some ...
21
votes
3answers
1k 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.
21
votes
1answer
209 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
21
votes
1answer
373 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: ...
19
votes
3answers
2k views

Find all the integral solutions to $x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$

If $x,y\in\mathbb{Z^+}$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried mod ...
19
votes
5answers
2k views

Can n! be a perfect square when n is an integer greater than 1?

Can n! be a perfect square when n is an integer greater than 1? (But is it possible, to prove without Bertrand's postulate. Because bertrands postulate is quite a strong result.)
19
votes
2answers
556 views

Does this system of simultaneous Pell-like equations have any non-trivial positive integer solutions?

Let $a,b,c$ be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is $(a,b,c)=(1,1,1)$. Are there others?
18
votes
2answers
377 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
2answers
390 views

Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?

Are there solutions in integers $a,b>1$ to the following simultaneous congruences? $$ a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2} $$ A brute-force search didn't turn up ...
18
votes
0answers
248 views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
18
votes
0answers
760 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (...
17
votes
2answers
1k 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 ...
17
votes
3answers
377 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.