Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

Use this tag for questions about finding integer, or perhaps rational, solutions to polynomial equations.

Diophantine equations are named after Diophantus of Alexandria, a third century Greek mathematician.

An example of a Diophantine equation is to find all quadruples of integers $(w,x,y,z)$ such that $$w^2+x^2=3(y^2+z^2).$$

Solving Diophantine equations often involves other areas of mathematics such as congruences, linear algebra, inequalities, forms (e.g., binary quadratic forms), and elliptic curves. Special solution methods include comparing divisors, considering orders of magnitude, Fermat's method of descent, and finding intersections of curves with lines of rational slope through a known rational point.

5324 questions
3
votes
3 answers

Solve modular arithmetic equation $\frac{1}{24} \cdot n(n+1)(n+2)(n+3) \equiv 1 \pmod{10}$

From another problem, I have reduced it to: This is the last step in solving: Solve $\frac{1}{24} \cdot n(n+1)(n+2)(n+3) \equiv 1 \pmod{10}$ How should I begin, a major problem is the $1/24$
Amad27
  • 10,465
3
votes
1 answer

how to solve this equation with the unknown at the powers

Please help me to solve this equation: Find $n \in \mathbb{N}$ such that: $\sqrt{1+5^n+6^n+11^n} \in \mathbb{N}$. $0$ is a particular solution, and does it have other one ?
hachemy
  • 314
3
votes
4 answers

Equation - what first?

I have this equation: $$ (x+y)(x^x + y^y) = 2009. $$ I must designate all pairs of integers satisfying the equation. What first? I tried multiply brackets , but to no avail
JWa
  • 59
3
votes
3 answers

All integer solutions to diophantine equation: $x^2+p y^2=z^2$?

I would like to find all integer solutions to the Diophantine equation $$ x^2+p y^2=z^2 $$ where $p\ge2$ is a given prime number. Also prove that my (probably parametric similar to Pythagorean triples $x=a^2-b^2,\ y=2ab,\ z=a^2+b^2$) solution gives…
asad
  • 929
3
votes
3 answers

Integer solutions for $x^2+y^2=208$

Which steps I can follow to find the integer solutions for the equation $x^2 + y^2 = 208$?
Mateus Coutinho Marim
  • 47
3
votes
1 answer

Integer solutions to $ n^2 + 1 = 2 \times 5^m$

What are the integer solutions to the diophantine equation $$n^2 + 1 = 2 \times 5 ^m? $$ We have $(n,m) = (3,1), (7, 2) $ as solutions. Are there any more? This seems like it would be a well known diophantine equation, but I can't seem to find any…
Calvin Lin
  • 68,864
3
votes
2 answers

Solving Diophantine equation $1/x^2+1/y^2=1/z^2$

How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation: $$1/x^2+1/y^2=1/z^2$$ Can we conclude that $x$ and $y$ are not coprimes for it to have solution?
Afonso Henriques
  • 223
3
votes
2 answers

Find all integers that make this expression rational

I came up with this difficult problem a while ago while solving another relatively easy problem. Find all integers m and n, such that $m^2 + n^2$ is a square, and such that $\sqrt{\frac{2m^2+2}{n^2+1}}$ is rational. I've already tried the…
Faraz Masroor
  • 1,210
3
votes
1 answer

Integer solution for $Rx^2+Sy^2=1$ .

Is there any integer solution in-terms of $R,S$ for the equation $Rx^2+Sy^2=1$ , . For example $(\frac{1}{\sqrt {2R}},\frac{1}{\sqrt {2S}})$ is a solution but not integer solution . Is there any integer solution tuple for the equation in terms of R…
hanugm
  • 2,353
  • 1
  • 13
  • 34
2
votes
3 answers

Integer solutions to $k^2m^2 -k^2 - m^2 +1 = n^2$

Can the positive integer solutions to $$ k^2m^2 -k^2 - m^2 +1 = n^2 $$ be characterized (in the sense that the solutions to $a^2+b^2 = c^2$ are characterized by $a=r^2-s^2, b=2rs, c=r^2+s^2$ with $(r,s)=1$ and $r\neq s \mod 2$)? Beyond than the…
Mark Fischler
  • 41,743
2
votes
1 answer

The Diophantine equation $x^p - 4y^p = z^2$ with $(x, y) = 1$ and $x, y, z > p.$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that $(x, y) = 1$ and $$x^{p} - 4y^{p} = z^{2}$$
Yes
  • 20,719
2
votes
0 answers

The Diophantine equation $x^n - y^n = z^2$

Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$ The question is: Does DMT apply to the equation $x^n - y^n = z^2$?
Yes
  • 20,719
2
votes
2 answers

Integer points of a circumference which radius in $n^{3/2}$

The question is: with a fixed integer $n$, what are the points with integer coordinates $(a,b)$ so that $a^2 + b^2 = n^3$? The equation is symmetric in $a$ and $b$, so if $(x,y)$ is a solution, then $(y,x)$ is a solution as well. Obviously if $n$ is…
ugosugo
  • 91
2
votes
1 answer

A diophantine equation

I'm trying to determine the ideal class group of $\mathbb{Q}(\sqrt{223})$ using elementary methods. Is there an easy way to show that $a^2 - 223b^2 = - 3$ has no integer solutions? I've tried reducing mod just about everything I could think of, but…
user18861
  • 23
2
votes
1 answer

How to solve equations like $x^2+y^2=2004^{2005}$?

I've found this kind of equation but I think I haven't enough mathematical tools to solve it. What would you do? $$x^2+y^2=2004^{2005}$$ Another kind: $$x^2+y^2=2005^{2004}$$
Surfer on the fall
  • 801
1 2 3
…
38 39