Product of distances

Number Theory Level 4

Let $a_1,a_2,a_3,a_4$ be distinct integers.

If $\displaystyle\prod _ {1\le i < j\le4} {|a_i - a_j|}$ is a perfect square,

then what is the minimum value of $\displaystyle\sum _ {1\le i < j\le4} {|a_i - a_j|} ?$

1 solution

Haosen Chen
Jul 19, 2018

WLOG $0=a_{1}<a_{2}<a_{3}<a_{4}$ ,since we only care about the difference between them.

Then the condition becomes that $a_{2}a_{3}a_{4}(a_{3}-a_{2})(a_{4}-a_{2})(a_{4}-a_{3})$ is a perfect square,

and what we want to minimize becomes $f=3a_{4}+a_{3}-a_{2}$ .

It's easy to find that $(a_{2},a_{3},a_{4})=(3,5,8)$ satisfies the condition and gives us $f=26$ .

When $a_{4}\ge 9$ ,we have $f\ge 3a_{4}+1\ge 28$ .So we only need to check the case when $a_{4}=3,4,5,6,7,8.$

Well,it's not hard.Just pay your patience to check one by one.

I guess that there might be a more elegant solution to this beautiful problem.So please share with us if you've found one.