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        riddles >> hard >> Integer Roots
        
(Message started by: Barukh on Feb 16^th, 2005, 3:34am)

Title: Integer Roots
Post by Barukh on Feb 16^th, 2005, 3:34am

Find three non-zero integers a, b, c such that the function f(x) = x³ + ax² + bx + c and its 1-st and 2-nd derivatives have all integer and distinct roots.

Title: Re: Integer Roots
Post by SMQ on Feb 18^th, 2005, 8:10am

One possible solution (of many) is :: a = -36, b = 285, c = -250 :: giving roots :: 1, 10, and 25 :: for the cubic, :: 5 and 19 :: for the first derivative, and :: 12 :: for the second derivative.

I started by working out formulas for a, b, and c in terms of the six roots, then made the simplifying assumption that there existed a solution where all the roots were positive. With that assumption I then wrote a program to iterate over all possible positive roots of the first derivative :: which uniquely determine the root of the second derivative :: searching for roots of the cubic :: which must sum to three times the root of the second derivative :: which, because of the assumption that all roots are positive is a finite search space. I gave the "smallest" solution above.

--SMQ

Title: Re: Integer Roots
Post by Barukh on Feb 19^th, 2005, 5:15am

Nicely done, SMQ and a nice start in the forum! :D

If you want, you can try to find the approach that is less computer-dependent.

Title: Re: Integer Roots
Post by SMQ on Feb 19^th, 2005, 6:51am

on 02/19/05 at 05:15:47, Barukh wrote:

If you want, you can try to find the approach that is less computer-dependent.

Ooh, discrete math and I have a long history of not getting along well. ;D I did derive some statements about the even/oddness and multiple-of-threeness of various groups of the roots, but it ended up being simpler not to use them in the program. I think for the moment I'll stick to doing continuous and/or infinite problems "by hand" and letting the machine handle discrete finite ones. ;)