Find the greatest positive integer

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The discussion focuses on finding the greatest positive integer \( x \) such that the expression \( x^3 + 4x^2 - 15x - 18 \) equals the cube of an integer. A key insight shared is that the expression is less than or equal to \( (x+1)^3 \), which helps in bounding the possible values of \( x \). Participants express appreciation for the collaborative problem-solving approach, highlighting the importance of recognizing mathematical inequalities in this context.

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Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.
 
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anemone said:
Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.
I had a slice of luck here
x must satisfy

$x^3 + 4x^2 - 15 x - 18 \le (x+1)^3$

or $x^2-18x - 19 \le 0$
so x = 19 makes the RHS 0

so x = 19 is the ans because we get a perfect square ( things would have been different had we not got integer)
 
Well done, kaliprasad!(Yes)

The trick is to recognize that the given expression is less than or equal to $(x+1)^3$.

Thanks for participating as well, my friend!:)
 

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