Determine all possible ordered triplets (A,B,C) of integers satisfying this simultaneous equation:

A³ + B³ + C³ = 3 and:
A + B + C = 3

Prove that there are no others.

Rewrite the second equation as A+B=3-C
Then the first as A³ + B³ = 3 - C³
factor and substitute
(A+B)(A²-AB+B²) =3 - C³
(3-C)(A²-AB+B²) =3 - C³
(A²-AB+B²) =(3-C³)/(3-C)
the RHS must be an integer, the division leads to
=C²+3C+9+ 24/(C-3)
which means (C-3) is a factor of 24 so C must be one of
{-21,-9,-5,-3,-1,0,1,2,4,5,6,7,9,11,15,26}
[And by the symmetry of A, B and C, they all must be one of these numbers.]

Put each of the above into (3-C³)/(3-C)
To create a new list of quotients Q={386,61,5,1,1,1,-5,61,61,71,89,121,166,281}
The two values that occur thrice: 1 and 61 are the only ones we need, they arise from permutations of A,B,C by the symmetry of these variables.

(A²-AB+B²)= Q
By the quadratic formula
A=.5B±.5√(4Q-3B²)

When Q=1, (C could be -1,0,or1)
the discriminant is only a perfect square if
B is -1,0,or1
leaving A to be one of -1,0,1 as well.
The only triple of these that actually works is {1,1,1}

When Q=61, (C could be -9,4 or 5)
B is ±9,±5,or±4
leaving A to be one of ±9,±5,or±4
The only triple of these that actually works is {4,4,-5}
and its permutations

Posted by Jer on 2014-03-22 23:48:51