Team selection test for the 54th IMO

Question

Find the least positive integer n, such that there exist positive integers a, b and c, none of which is a perfect square and a^3 + b^3 + c^3 - 3abc = 2013^n.

Accepted Answer

It is easy to be seen that the left hand side is divisible by

9

implying that the equation has no solution for

n = 1

.

Let

n = 2

. We have

201 3^{2} = (3 \cdot 11 \cdot 61)^{2}

,

3^{2} = 2^{3} + 1^{3}

and

11 \cdot 61 = 8^{3} + 4^{3} + 1^{3} + 3 \cdot 8 \cdot 4 \cdot 1

. We use the identity

(a^{3} + b^{3} + c^{3} - 3 ab c) (x^{3} + y^{3} + z^{3} - 3 x y z) = u^{3} + v^{3} + w^{3} - 3 uv w .

where

u = a x + b y + cz

,

v = a y + b z + c x

and

w = a z + b x + cy

. We apply the above identity first for

(8, 4, - 1)

and

(4, 8, - 1)

, and then for the obtained triples

(65, 0, 56)

and

(2, 1, 0)

and obtain

6 5^{3} + 5 6^{3} = (11 \cdot 61)^{2}

and

13 0^{3} + 17 7^{3} + 5 6^{3} - 3 \cdot 130 \cdot 177 \cdot 56 = 201 3^{2}

.