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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
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How many quadruples are there?

How many quadruples are there $(x_1,x_2,x_3,x_4) \in \mathbb{Z}^{+}_{0}$ such that $(x_1+x_2)(2x_2+2x_3+x_4)=95$? My attempt. We have that $x_1+x_2 = 19$ and $2x_2+2x_3+x_4 = 5$ or $x_1+x_2 = 5$ and $2x_2+2x_3+x_4 = 19$. In the first case, we have…
user19405892
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Prove that the equation $\ 5x^4 + x − 3 = 0\ $ has no rational solutions.

I'm locked at $\ x\left(5x^3 + 1\right) = 3$. Not too sure where to go from there but I'm getting the feeling it's really really obvious..
Rohan
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Positive rationals satisfying $a+b+c+d=abcd$?

Victor has posted a couple of problems involving finding real and rational solutions of $a+b+c=abc$. Two techniques have been given: using triangles, and using scaling. Neither seems to work for the following problem. How can one easily (without…
user13618
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Diophantine equation $x^3+z x^2-z y^2=0$

I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation $$ x^3+zx^2-zy^2=0 $$ where $x,y,z\in\mathbb{Z}$
Norbert
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A conjecture about Pythagorean triples

I noticed for the integer solutions of $a^2 + b^2 = c^2$, there don't seem to be cases where both a and b are odd numbers. In fact, I saw this property pop up on a nice question, which required you to prove it. So I have tried proving it, but I have…
Just_a_fool
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How can this equation be solved?

I have no idea how to solve this equation. $$x^2y^2+324y^2+64x^2-36xy^2-16x^2y+144xy = 0 $$ Sorry $x,y \in \mathbb{+Z}$
Veritas
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Can $ x^2-4a^n$ be a square?

Let $ x,y,a,n \in {Z}>1$, Does the diophantine equation $ x^2-4a^n=y^2$ have any solutions? or How can I solve this?
user97615
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2 answers

Solving $3 = x^2 + y^2 + z^2 - xy - yz - zx$ for integer $x$, $y$, $z$

I've been thinking about a solution for the following equation for integers $x, y, z$: $$3 = x^2 + y^2 + z^2 - xy - yz - zx$$ A possible approach would probably be to transform the original equation to the following one: $$6 = (x-y)^2 + (y-z)^2 +…
Michael Schreiber
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Quadratic equation with only one solution on integers

$$x^2-y^2=2xy $$ given this Diophantine equation is also possible to prove or disprove that it has no solution on integers except $ x=y=0$ ? Can it have rational numbers as a solution ?
Jose Garcia
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Why does the equation $a^4+3b^4=c^4+3d^4$ have solutions in such small integers?

The smallest non-trivial solution of the diophantine equation: $$a^4+b^4=c^4+d^4\qquad(1)$$ is $(a,b,c,d)=(59,158,133,134)$ (see here, equation 116). One might expect that integers of the form $m^4+3n^4$ are even more sparsely distributed than…
Adam Bailey
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Fermat's Last Theorem for $n=4$ revisited

Fermat's Last Theorem for $n=4$ states that If $$b^4+c^4=a^4\tag{1}$$ Then, no positive integers $a, b, c$ exist simultaneously. We can re-arrange (1) as $$a^4-b^4=c^4\tag{2}$$ But, by direct expansion, we see that $$a^4-b^4 =…
Shuaib Lateef
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Solve for integer values of $x,y,z$: $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$

Solve for integer values of $x,y,z$; $$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3.$$ My attempt: Note that all $x, y, z$ are non-zero, otherwise a denominator would be zero. Mutliplying by $xyz$ gives: $$ x^2y^2+x^2z^2+y^2z^2 = 3xyz $$ Since…
cyanide
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Find $x, y$, and $z$ such that $x^3+y^3+z^3=k$, for each $k$ from $1$ to $100.$

It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”. A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer…
user730361
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Find the least possible natural n such that $8x + 13y = n$ will have exactly 9 solutions in natural numbers

Well, to approach this, we firstly should understand what is $gcd$ of 8 and 13. It is 1, so we do not require $n$ to have any divisibility properties. Means that we did not shorten our first range at all. Of course we can do some "binary search"…
math-traveler
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Diophantine equation: $x^4+4=py^4$

find all primes p for which $x^4+4=py^4$ is solvable for integers. My try: I started out with $p=2$ . LHS must be even, thus $(x^4+4)\mod(16)=4$ similarly $(2y^4)\mod (16)=${$0,2$}. Thus for $p=2$ there are no solutions. Now I tried factoring i.e…
Albus Dumbledore
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