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anemone
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Determine all triplets $(x,\;y,\:z)$ of positive integers such that $x \le y \le z$ and
$x+y+z+xy+yz+zx=xyz+1$.
$x+y+z+xy+yz+zx=xyz+1$.
anemone said:Determine all triplets $(x,\;y,\:z)$ of positive integers such that $x \le y \le z$ and
$x+y+z+xy+yz+zx=xyz+1$.
mente oscura said:Hello.
[tex](x+1)(y+1)(z+1)=2(1+xyz)[/tex]
At a glance:
[tex](1,0,0) \ (0,1,0) \ (0,0,1) \ (-1,-1,-1)[/tex]
Only meets the two restrictions: [tex](1,0,0)[/tex]
Regards.
eddybob123 said:The above solution is not correct. The question asks for positive integers, and $0$ IS NOT POSITIVE!
$$2,4,13$$
$$2,5,8$$
$$3,3,7$$
[sp]If $x$ (the smallest of these numbers) is $\geqslant4$ then each of $x,y,z$ is $\leqslant\frac14yz$; and each of $xy,xz,yz$ is $\leqslant yz$. Therefore $x+y+z+xy+yz+zx \leqslant \frac{15}4yz$. But $xyz + 1>4yz > \frac{15}4yz$. So there cannot be any soluions with $x\geqslant4.$anemone said:Determine all triplets $(x,\;y,\:z)$ of positive integers such that $x \le y \le z$ and
$x+y+z+xy+yz+zx=xyz+1$.
The equation is represented as $x+y+z+xy+yz+zx=xyz+1$ in mathematical notation. It is also known as a Diophantine equation, where all variables are integers and the equation must be satisfied for a specific set of values.
The solutions to a Diophantine equation can be found through various methods, such as trial and error, substitution, or using modular arithmetic. In this specific case, we can use the fact that all integers have a unique prime factorization to find the solutions.
Yes, there are some general strategies that can be applied to solving Diophantine equations, such as factoring, finding common divisors, or using congruence relations. However, each equation may require a different approach, and there is no one-size-fits-all method.
No, some Diophantine equations have no integer solutions. This is known as an unsolvable equation. For example, the equation $x^2+y^2=3$ has no integer solutions.
Yes, Diophantine equations have many real-world applications, especially in fields such as number theory and cryptography. In this specific equation, finding solutions can also be used in computer algorithms and coding theory.