If $n$ is a positive integer greater than 1 such that $3n+1$ is perfect square, then show that $n+1$ is the sum of three perfect squares.
My work:
$3n+1=x^2$
$3n+3=x^2+2$
$3(n+1)=x^2+2$
$(n+1)=\dfrac{x^2+2}{3}$
I have no clue what to do next. Please help!
Best Answer
We are given that $3n+1 = a^2 $.
We want to show that $n+1$ is the sum of 3 perfect squares.
Note that $a$ is not a multiple of 3.
If $ a \equiv 1 \pmod{3}$, then observe that $9n+9 = 3a^2 + 6 = (a-1)^2 + (a-1)^2 + (a+2)^2$, and hence
$$ n+1 = \left( \frac{a-1}{3} \right)^2 + \left( \frac{a-1}{3} \right)^2 + \left( \frac{a+2}{3} \right)^2. $$
If $ a \equiv 2 \pmod{3}$, then observe that $9n+9 = 3a^2 + 6 = (a+1)^2 + (a+1)^2 + (a-2)^2$, and hence
$$ n+1 = \left( \frac{a+1}{3} \right)^2 + \left( \frac{a+1}{3} \right)^2 + \left( \frac{a-2}{3} \right)^2 $$
Thus the result is true.
The motivation behind the solution is: We want to show that $n+1$ is the sum of 3 squares, and the only thing that we have to work with is $a^2$, and possibly things around it. Since $3a^2$ (the naive sum of 3 squares) is so close to $9n+9$, this suggests that we have some sort of wriggle room. Remembering that we have to account for the factor of 3 then greatly restricts our options.
As an extension, show that $n+3$ can also be written as the sum of 3 perfect squares, using a similar approach.