2016 JBMO Problems/Problem 3

Problem

Find all triplets of integers $(a,b,c)$ such that the number

is a power of $2016$ .

(A power of $2016$ is an integer of form $2016^n$ ,where $n$ is a non-negative integer.)

Solution

It is given that $a,b,c \in \mathbb{Z}$

Let $(a-b) = -x$ and $(b-c)=-y$ then $(c-a) = x+y$ and $x,y \in \mathbb{Z}$

$(a-b)(b-c)(c-a) = xy(x+y)$

We can then distinguish between two cases:

Case 1: If $n=0$

$2016^n = 2016^0 = 1 \equiv 1 (\mod 2016)$

$xy(x+y) = -2$

$-2 = (-1)(-1)(+2) = (-1)(+1)(+2)=(+1)(+1)(-2)$

$(x,y) \in \{(-1,-1), (-2,1), (-1,2)\}$

$(a,b,c) = (k, k+1, k+2)$ and all cyclic permutations, with $k \in \mathbb{Z}$

Case 2: If $n>0$

$2016^n \equiv 0 (\mod 2016)$

$xy(x+y) + 4 = 2 \cdot 2016^n$

$2016 = 2^5 \cdot 3^2 \cdot 7$ is the unique prime factorization.

Using module arithmetic, it can be proved that there is no solution.

Method 1: Using modulo 9

$xy(x+y) + 4 \equiv 0 (\mod 9)$

$xy(x+y) \equiv 0 (\mod 9)$

In general, it is impossible to find integer values for $x,y$ to satisfy the last statement.

One way to show this is by checking all 27 possible cases modulo 9. An other one is the following:

$xy(x+y) + 4 \equiv 0 (\mod 3)$

$xy(x+y) \equiv 0 (\mod 3)$

$x \equiv 1 (\mod 3)$ and $y \equiv 1 (\mod 3)$

$x,y \in \{ 1(\mod 3), 4(\mod 3), 7(\mod 3) \}$

$xy(x+y) \equiv 2 (\mod 9)$ which is absurd since $xy(x+y) \equiv 5 (\mod 9)$ .

Method 2: Using modulo 7

$xy(x+y) + 4 \equiv 0 (\mod 7)$

$xy(x+y) \equiv 3 (\mod 7)$

In general, it is impossible to find integer values for $x,y$ to satisfy the last statement.

One way to show this is by checking all 15 possible cases modulo 7.

Solution 2

First, we check when $n>0$ . WLOG, let $a>b$ since one of the factors of $(a-b)(b-c)(c-a)$ must be positive. Then, $a>c>b$ is forced. Let $a-c=d_1$ and $c-b=d_2$ so we want $2(2016^n-2)=4(1008^{2n-1}-1)=d_1d_2(d_1+d_2)$ . Now, we have two cases, one of $d_1$ or $d_2$ is odd and the other is even, or they are both odd and $d_1+d_2\equiv 0\pmod{4}$ . Case 1: WLOG $d_1$ is odd and $d_2$ is even. Then $d_2\equiv 0\pmod{4}$ . Let $d_2=4s$ where $s$ is odd. Note that $s$ cannot be even since the product cannot be divisible by 8. After dividing the modular equation by 4, we get $1008^o\equiv 1\pmod {(4s+d_1)(sd_1)}, o=2n-1$ . Remember that Fermat's LIttle Theorem states that $a^{p-1}\equiv 1\pmod{p}$ if $p\nmid{a}$ . There must be a prime $p$ that divides $(4s+d_1)(sd_1)$ , and by CRT, that means that $o$ is one less than that $p$ . However, $o$ must be odd so the only $p$ that works is $p=2$ . Thus, this implies that $(4s+d_1)(sd_1)$ has no divisor of any other prime other than 2, so it must be a power of 2. However, since $d_1$ is odd, it cannot have a factor of 2. We arrive at a contradiction.