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
0
votes
2 answers

Integer Solutions Explaination

I know how to solve linear diophantine equations, but I was wondering if someone can give me a step by step to solve something like $2x^2 + 2x - 5y = -1$? I cannot find a lot of resources on this particular form. I know the solutions are $${ y = 10…
Ryan Topps
  • 27
0
votes
3 answers

Area of the polygon with coordinates that satisfy the equation

Determine the area of the polygon formed by the ordered pairs (x, y) where x and y are positive integers which satisfy the equation $\frac{1}{x} + \frac{1}{y}= \frac{1}{13}$ I got : $(x-13)(y-13)=169$ but it only has 2 solutions with positive…
SuperMage1
  • 2,486
0
votes
1 answer

Elementary solution to the diophantine equation $n(n+1)=4m(m+1)$?

Recently I tried to solve a diophantine equation $n(n+1)=4m(m+1)$ with $n,m\in\mathbb{Z}$ which resulted from an other equation. But how can one show, that there are no non-trivial solutions. Obvsiously there are four solutions. But is there an…
Cornman
  • 11,065
  • 4
  • 30
  • 57
0
votes
1 answer

How would I approach this diophantine equation with powers of two? $(6k-1)2^{2(n-1)} = 9l-1$

A friend of mine recently gave me the follwing diophantine equation ($k, l, etc. \in \Bbb N$) and asked me whether I could solve it, i.e. "find all solutions" (I'm still a high school student, so please bear with me and please suggest improvements…
Moritz
  • 101
0
votes
2 answers

The Diophantine equation $x_1^6+x_2^6+y^6=z^2$ where both $(x_i)\equiv 0{\pmod 7}$.

The Diophantine equation $x_1^6+x_2^6+y^6=z^2$ where both $(x_i)\equiv 0{\pmod 7}$. As a logical follow on to The Diophantine equation $x_1^6+x_2^6+x_3^6=z^2$ where exactly one $(x_i)\equiv 0{\pmod 7}$. I have considered the case with exactly two of…
Old Peter
  • 1,357
0
votes
2 answers

How can Falting's theorem be used to decide whether infinite many rational solutions exist?

Falting's theorem states that a non-singular algebraic curve with genus $g>1$ only has finite many rational points. Apparently, the degree-formula (see https://en.wikipedia.org/wiki/Genus%E2%80%93degree_formula) allows to determine the genus if the…
Peter
  • 84,454
0
votes
1 answer

How to solve the diophantine equation $x^2-x=y^3$

How would I solve the diophantine equation $x^2-x=y^3$? Would anyone be able to help solve this equation? We have only done equations up to powers of $2$ as of now.
568075260
  • 13
0
votes
2 answers

Problem with Diophantine equations

A man purchases an item for 10.08 and instead of paying cash he tendered a cheque which he had with him. the cashier mistook the number of cents for the number of dollars and the number of dollars for the number of cents and gave back 7 times the…
Minura
  • 13
0
votes
0 answers

Information on Taxicab$(n,2,2)$

Taxicab$(n,2,2)$ is the smallest number expressible as the sum of 2 $n$th powers in 2 different ways. I believe only the first 4 are known: Taxicab$(1,2,2) = 4 = 1^1 + 3^1 = 2^1 + 2^1$, Taxicab$(2,2,2) = 50 = 1^2 + 7^2 = 5^2 +…
Supware
  • 940
0
votes
1 answer

Help solving the system below

I am trying to solve the system of diophantine equations below where $x,y$ are coprime \begin{eqnarray} xr^2+ys^2&=&uva\\ xr+ys&= &uvb\\ x+y&=&uc \end{eqnarray} I solve 2 out of 3 then replace in the remaining one. However, things get messy at…
user97615
0
votes
0 answers

Linear Diophantine Equation vs linear congruence

I just saw the next two theorems and I asked myself some questions about it. Linear Diophantine Equation: Let a and b be integers with $d=(a,b)$. The equation $ax+by=c$ has: 1) No integral solutions if $c$ is not divisible by $d$ 2) Infinity many…
WinstonCherf
  • 1,022
0
votes
1 answer

Solve the system of diophantine equations: $xz=a(a+b)$; $xt+yz=2ab^2$; $yt=b^4$.

I am trying to solve the following system of diophantine equations for $x,y,z,t$ where $\gcd(a,b)=1$ are coprime: $$xz=a(a+b)$$ $$xt+yz=2ab^2$$ $$yt=b^4$$ I noticed that $x,z$ share a common factor and $y,t$ have the same set of primes.…
user97615
0
votes
1 answer

Diophantine equation - all solutions

I have a question concerning diophantine equations. I get the first steps right with the Euclides Algorithm and then doing it backwards. However, when trying to find all solutions I often make some small mistakes so I figured that I don't understand…
lykkinski
  • 93
0
votes
3 answers

How to find $x$ such that $a^2-b^2-(c+d)x=r^2$; $c^2-d^2+(a+b)x=s^2$

Let $a,b,c,d,r,s,x$ be integers. Find all integers $x$ such that the following quantities are integral squares: $$a^2-b^2-(c+d)x=r^2$$ $$c^2-d^2+(a+b)x=s^2$$ First, I noticed that if $$(a+b)(c+d)=0$$ then it is easy to solve. I supposed…
user97615
0
votes
1 answer

Find all the integer solutions of $x^2-y^2=a^2+mb^2$

Let $x,y,a,b,m$ be non-zero integers where $m$ is square free. I am trying to find all the integer solutions of $$x^2-y^2=a^2+mb^2$$ I tried to factorize both sides but the righthand side equation is not too friendly. Any hints?
user97615
1 2 3
…
38 39