Questions on finding integer/rational solutions of equations.

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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 ...
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 $x^y = y^x$?
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: $...
3
votes
3answers
2k 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 ...
12
votes
6answers
4k 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 ...
-1
votes
4answers
356 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 ...
14
votes
5answers
1k 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 $x^2+y^2=z^2?$
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 ...
17
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. ...
9
votes
3answers
1k views

Erdős-Straus conjecture

I'm reading a lot about the Erdős-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 ...
1
vote
5answers
526 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
6answers
640 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 ...
1
vote
5answers
789 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.
14
votes
6answers
1k 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 $$\frac{1}{2}n(3n-1)=\frac{1}...
3
votes
3answers
1k views

How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...
13
votes
5answers
902 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 ...
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, ...
11
votes
2answers
516 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 read from the book "About Indeterminate Equation" (in Chinese, by ...
11
votes
5answers
2k 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 ...
6
votes
6answers
3k 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 ...
4
votes
2answers
598 views

Finding all solutions of the Pell-type equation $x^2-5y^2 = -4$

I wanted to solve the equation $x^2-5y^2 = -4$ with $x$ and $y$ integers. Let $\omega=\frac{1+\sqrt5}{2}$ and $A = \mathbb{Z}[\omega]$. One can reduce the Pell equation to finding the elements of $A$ ...
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.)
8
votes
3answers
722 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
1answer
278 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 c^...
2
votes
1answer
226 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 ...
6
votes
5answers
421 views

Find all integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$

Characterize all positive integers $x$, $y$, and $z$ such that: $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$$ For example, $\dfrac{1}{x+1} + \dfrac{1}{x(x+1)} = \dfrac{1}{x}$.
6
votes
3answers
2k 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
246 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
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 (...
3
votes
6answers
540 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
2
votes
5answers
315 views

Solving a Diophantine equation of the form $x^2 = ay^2 + byz + cz^2$ with the constants $a, b, c$ given and $x,y,z$ positive integers

Is there any procedure for determining if an infinite amount of solutions exist for an equation of the type $x^2 = ay^2 + byz + cz^2$ for arbitrary integer constants $a, b, c$ and variables $x, y, z \...
2
votes
2answers
243 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
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 $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
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? ...
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 ...
8
votes
3answers
927 views

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

How does one find all positive integer solutions to the equation $x^3+48=y^4$?
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 ...
16
votes
3answers
1k views

Machin's formula and cousins

There exists a well-known formula by John Machin: $$\frac{\pi}{4} = 4 \arctan \left(\frac{1}{5}\right) - \arctan \left(\frac{1}{239}\right).$$ Actually, it belongs to the family of Machin-like ...
16
votes
2answers
329 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, ...
15
votes
7answers
3k views
8
votes
1answer
358 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 N....
12
votes
2answers
789 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, ...
7
votes
1answer
528 views

The sum of the first $n$ squares is a square: a system of two Pell-type-equations

This question comes from trying to see why 24 is the only non-trivial value of $n$ for which $$1^2+2^2+3^2+\cdots+n^2$$ is a perfect square. To this end, let $m,n \in \mathbb N$ be such that $1^2+2^2+...
10
votes
3answers
1k views

Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$

Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$. My Attempt: We will calculate for $x,y,z>0$. Then, using the AM-GM Inequality, we have $$ \begin{...
7
votes
2answers
1k 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
5answers
233 views

Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers?

Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct integer values with $x,y$ positive integers. In other words, there ...
1
vote
1answer
180 views

Parametrization of solutions of diophantine equation

The issue I discussed in this thread. Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$ Generally speaking at the forum often ask a question about this equation. So I ...
1
vote
2answers
609 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.
6
votes
3answers
399 views

Whenever Pell's equation proof is solvable, it has infinitely many solutions

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 -cy^...
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 ...