Integral solutions only!

Algebra Level 4

∣ y 3 + 1 y 2 z y 2 x y z 2 z 3 + 1 z 2 x y x 2 x 2 z x 3 + 1 ∣ = 11 \begin{vmatrix} { y }^{ 3 }+1 & { y }^{ 2 }z & { y }^{ 2 }x \\ y{ z }^{ 2 } & { z }^{ 3 }+1 & { z }^{ 2 }x \\ y{ x }^{ 2 } & { x }^{ 2 }z & { x }^{ 3 }+1 \end{vmatrix}=11

What is the number of positive integral solutions of the equation above?


The answer is 3.

Utkarsh Bansal by Utkarsh Bansal , 23, India

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2 solutions

Bill Bell
Feb 25, 2015

I used the Python sympy module to calculate the determinant.

It yields

x 3 + y 3 + z**3 + 1

Thus, we have the Diophantine equation

x 3 + y 3 + z 3 = 10 { x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }=10

Only three cubes add to 10: 1, 1 and 8. There are three permutations of these numbers.

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Nice solution. thank you sir.

Utkarsh Bansal - 6 years, 3 months ago

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Thank you, and thank you for posing the problem.

Bill Bell - 6 years, 3 months ago

multiply each row with y , z , x y,z,x respectively, and now take y , z , x y,z,x common from each column , then we get simplified version to solve the determinant , which will turn out to be

x 3 + y 3 + z 3 x^3+y^3+z^3 = 10,

So positive solutions are (1,1,2) , (1,2,1) , (2,1,1).

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