Crocodile System

Solve for positive integers x, y & z :

x+yz=4x + y - z = 4

x2y2+z2=4x^2 - y^2 + z^2 = 4

xyz=6xyz = 6

Note by Dev Sharma
4 years, 3 months ago

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Usual Method:

There're no integral solutions. As (x,y,z)(x,y,z) will only be the 9 permutations of (1,1,6)(1,1,6) and (1,2,3)(1,2,3). None of them satisfy.


Brute Force Method:

I also did some nonsense calculations to find the values:

x+y=4+z,xy=6zx+y = 4+z, xy=\dfrac{6}{z}

Thus I obtained xy=z3+8z2+16z24zx-y = \sqrt{\dfrac{z^3+8z^2+16z-24}{z}}

(x+y)(xy)=4z2(x+y)2(xy)2=(4z2)2(x+y)(x-y) = 4 - z^2 \Rightarrow (x+y)^2(x-y)^2 = (4 - z^2)^2

(4+z)2(z3+8z2+16z24z)=(4z2)2\Rightarrow (4+z)^2 \left(\dfrac{z^3+8z^2+16z-24}{z} \right)= (4 - z^2)^2

And then visit this.

No integral solutions obtained by Brute Force as well.

Satyajit Mohanty - 4 years, 2 months ago

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Thanks sir.

But how do you find x - y

Dev Sharma - 4 years, 2 months ago

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(xy)2=(x+y)24xy(x-y)^2 = (x+y)^2 - 4xy

Satyajit Mohanty - 4 years, 2 months ago

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@Dev Sharma : Is this any of NMTC's problems? Nowadays, NMTC is trending among the young students of India!

Satyajit Mohanty - 4 years, 2 months ago

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@Satyajit Mohanty sir, it RMO problem. What is full form of NMTC?

Dev Sharma - 4 years, 2 months ago

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@Dev Sharma National Mathematics Talent Contests

Satyajit Mohanty - 4 years, 2 months ago

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@Satyajit Mohanty Sir, are you professor?

Dev Sharma - 4 years, 2 months ago

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@Dev Sharma As you can see, I'm only 18 years old. So, I'm a student, not a professor :D And please don't call me Sir. Sounds weird :/

Satyajit Mohanty - 4 years, 2 months ago

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I made it to a bi quadratic in z but no integral sol.

Aakash Khandelwal - 4 years, 2 months ago

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