Find four distinct positive integers such that the sum of any three of them is a perfect square.

I created a spread sheet to calculate the positive integers based on a set of 4 perfect squares. The equation to calculate the integers based on the set of squares is...
I1 = (S2+S3+S4-2*S1)/3
I2 = (S1+S3+S4-2*S2)/3
I3 = (S1+S2+S4-2*S3)/3
I4 = (S1+S2+S3-2*S4)/3

Where I1 is the smallest integer and I4 is the largest integer in the solution, and S1 is the smallest square and S4 is the largest square.

Using this I can count the soultions for any set of squares with a given maximum.

The first set I found with a solution had 11^2 as it's largest member. Here's a list of the maximum square and the number of solutions...

Square Solutions
11^2 ---- 1
12^2 ---- 2
13^2 ---- 4
14^2 ---- 7
15^2 ---- 11
16^2 ---- 17
17^2 ---- 25
18^2 ---- 36
19^2 ---- 49
20^2 ---- 66
21^2 ---- 85
22^2 ---- 110

Each set includes the solutions of the previos set. I don't know if there is a way to describe the complete set of solutions, but I could list plenty of solutions.

Posted by Robert on 2005-02-24 06:28:48