Questions on finding integer/rational solutions of polynomial equations.

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50
votes
6answers
4k 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 $x^y = y^x$?
17
votes
5answers
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 ...
2
votes
3answers
934 views

Solutions to $ax^2 + by^2 = cz^2$

The integer solutions to the equation $x^2 + y^2 = z^2$ are very well studied. I'm wondering if there's any literature about the integer solutions to the equation $ax^2 + by^2 = cz^2$ where a,b,c are ...
26
votes
3answers
21k 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: ...
9
votes
4answers
3k views

Diophantine equation $a^2+b^2=c^2+d^2$

I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation $$a^2+b^2=c^2+d^2$$ I tried a web search and found nothing on this one. I'm trying to avoid ...
13
votes
5answers
941 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 $x^2+y^2=z^2?$
5
votes
3answers
862 views

Erdös-Straus conjecture

I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...
17
votes
3answers
2k 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 ...
3
votes
6answers
446 views

Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
0
votes
4answers
225 views

Curves triangular numbers.

Sometimes you have to deal with this equation: $X^2+aX+Y^2+bY=Z^2+cZ$ $a,b,c$ - integer coefficients. I wrote below - to start a particular solution of Diophantine equations. To do this, use the ...
4
votes
5answers
2k views

Find all solutions: $x^2 + 2y^2 = z^2$

I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem: Find all solutions in ...
11
votes
2answers
2k 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: ...
7
votes
3answers
658 views

When do the multiples of two primes span all large enough natural numbers?

It is well-known that given two primes $p$ and $q$, $pZ + qZ = Z$ where $Z$ stands for all integers. It seems to me that the set of natural number multiples, i.e. $pN + qN$ also span all natural ...
1
vote
5answers
458 views

Number of solution for $xy +yz + zx = N$

Is there a way to find number of "different" solutions to the equation $xy +yz + zx = N$, given the value of $N$. Note: $x,y,z$ can have only non-negative values.
2
votes
2answers
4k views

How many integer solutions to a linear combination, with restrictions?

I've already done a few problems such as this, other problems where I'm supposed to find the number of combinations or permutations, subject to certain restrictions. Here's been my basic strategy: ...
4
votes
1answer
541 views

How would you solve the diophantine $x^4+y^4=2z^2$

I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out. By solving I mean fining all solutions and proving there are no more. Keith ...
9
votes
3answers
393 views

Solving the diophantine equation $y^{2}=x^{3}-2$

It is known that the diophantine equation $y^{2}=x^{3}-2$ has only one positive integer solution $(x,y)=(3,5)$. The proof of it can be seen from the book "About Indeterminate Equation (in Chinese, by ...
10
votes
4answers
1k 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 ...
7
votes
2answers
986 views

Solve the Diophantine equation $ 3x^2 - 2y^2 =1 $

Solve $$ 3x^2 - 2y^2 =1 $$ in $ \mathbb{Z}$. How can we do it? ( All of answers gave me a great help. Thanks a lot kind stackexchangers.)
6
votes
3answers
934 views

Derivation of Pythagorean Triple General Solution Starting Point:

I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation: $$ x^2 + y^2 = z^2, $$ where $x,y,z > 0$ are integers. I came across the ...
2
votes
2answers
221 views

integer solutions of $xy+9(x+y)=2006$

How many integer solutions does $xy+9(x+y)=2006$ have ? Here x and y are both integers . My trying : I have tried to solve this problem But I have no idea to solve this . Please help
1
vote
1answer
235 views

How to prove Greatest Common Divisor using Bézout's Lemma

The problem is to prove the following If $\gcd(a,b) = c$, then $\gcd(a^m, b^m) = c^m$ I know that this can be solved easily by proving that $c\mid a \implies c^m \mid a^m$ and $c\mid b \implies ...
2
votes
1answer
129 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
12
votes
0answers
620 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 ...
26
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 ...
40
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 $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
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? ...
26
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 ...
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 ...
8
votes
3answers
823 views

$x^3+48=y^4$ does not have integer (?) solutions

How does one find all positive integer solutions to the equation $x^3+48=y^4$?
8
votes
1answer
345 views

On the Diophantine equation $a^2+b^2 = c^2+k$

Given the Diophantine equation, $$a^2+b^2=c^2+k$$ where k is a constant integer. Let $0 < a \le b$, and $\Delta_k(N)$ be the number of primitive solutions with $0 < c < N$ for some bound ...
12
votes
2answers
545 views

Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$. It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd. Assuming $p = q + 2k$ we conclude, by the equation, ...
4
votes
1answer
732 views

The Diophantine equation $x^2 + 2 = y^3$

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)
4
votes
3answers
587 views

Natural number solutions to $\frac{xy}{x+y}=n$ (equivalent to $\frac 1x+\frac 1y=\frac 1n$)

I have a question about the following problem from a Putnam review: Let $n\in \mathbb{N}$. Find how many pairs of natural numbers $(x, y)\in \mathbb{N}\times \mathbb{N}$ solve $$ \frac{xy}{x+y}=n. $$ ...
1
vote
2answers
483 views

Generating Pythagorean triples for $a^2+b^2=5c^2$?

Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.
7
votes
4answers
2k views

General formula to obtain triangular-square numbers

I am trying to find a general formula for triangular square numbers. I have calculated some terms of the triangular-square sequence ($TS_n$): $TS_n=$1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ...
2
votes
3answers
231 views

Sum of roots is integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
46
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 ...
25
votes
1answer
2k 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 ...
16
votes
2answers
300 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, ...
17
votes
1answer
528 views

More elliptic curves for $x^4+y^4+z^4 = 1$?

(Note: This has been updated to be similar with this MO post.) There are exactly 22 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{11}$. Noam Elkies showed that $(1)$ as, ...
13
votes
7answers
3k views
13
votes
2answers
632 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 ...
8
votes
5answers
582 views

Proof that the equation $x^2 - 3y^2 = 1$ has infinite solutions for $x$ and $y$ being integers

I have seen the Pell's equation wiki page but I need to prove this from scratch without mentioning any formula. I have also seen multiple answers on this site but the answers tend to skip over and ...
5
votes
2answers
4k views

Count the number of positive solutions for a linear diophantine equation

Given a linear Diophantine equation, how can I count the number of positive solutions? More specifically, I am interested in the number of positive solutions for the following linear Diophantine ...
16
votes
2answers
1k views

Find all integer solutions for the equation $|5x^2 - y^2| = 4$

In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation $$|5x^2 - y^2| = 4$$ occur when $x$ is a Fibonacci number and $y$ is a Lucas number. ...
10
votes
3answers
301 views

Every natural number is representable as $\sum_{k=1}^{n} \pm k^5$ … if somebody proves it for 240 integers

(This post is inspired by "Is every $\mathbb{N}$ representable as $\sum\limits_{k=1}^{n} \pm k^3$"? My question is at the end.) The problem of whether every natural number $N$ is, $$N=\sum_{k=1}^n ...
7
votes
3answers
745 views

Integer Solutions to $x^2+y^2=5z^2$

I'm looking for a formula to generate all solutions $x$, $y$, $z$ for $x^2 + y^2 = 5z^2$. Any advice?
6
votes
1answer
2k views

How to find an integer solution for general Diophantine equation ax + by + cz + dt… = N

I know how to solve ax + by = c using Extended Algorithm. But with more than variables, I'm lost :(. To verify if it has an integer solution is easy, since we only ...
3
votes
3answers
195 views

Positive integral solutions of $3^x+4^y=5^z$

Are there more integral solutions for $3^x+4^y=5^z$, than $x=y=z=2$ ? If not, how do I show that? I could show that for $3^x+4^x=5^x$, but I'm stuck at the general case? Any ideas, maybe graphs, ...