$a^2+b^2+c^2=nd^2$

Question

This from the $2022$ Francophone Mathematics Olympiad:

Find the smallest natural number $n$ such that the only solution to: $a^2+b^2+c^2=nd^2$ $(a,b,c,d$ are intgers$)$ is $(0,0,0,0)$

Here is my solution:

For $n≤6$, we can see that the condition is not satisfied. Let's show that $n=7$ is the desired one. Suppose not: if $d$ is odd then working mod $8$ shows that there are no solutions. If $d$ is even, then we let $(a_0,b_0,c_0,d_0)$ be the smallest solution in which one of $a_0,b_0,c_0$ is not zero. Then working mod $4$ tells us that $a_0,b_0,c_0$ must all be even. And we have a smaller solution $(a_0/2,b_0/2,c_0/2,d_0/2)$.

Are there any other approaches different than mine?

Answer 1

Leonard E. Dickson Modern Elementary Theory of Numbers 1939

from pages 11, 112, 113