What is the largest set possible such that every 3 sum to a square?

I do not know the answer but there is a set of 4 numbers that I know.

{1,22,41,58} any three of these is a square.

For your first question, it is an open problem whether there is a set of six positive integers that pairwise are a square.

]]>If a,b and c are odd, a+b, b+c and a+c are even, and must be multiples of 4 since they are square. But a+b+c is odd, and a+b+csad(a+b)+(b+c)+(a+c))/2 which is even. Therefore at most 2 of the integers can be odd.

If a is odd, and the rest are even, a+b,b+c≡1(mod4), b+c≡2-a(mod4), a≡2(mod4).

If a and b are odd, and the rest are even, a+b≡0(mod4), a+c,b+c≡1(mod4), a+2c+b≡2(mod4), c≡1(mod4) but c is meant to be even.

The only remaining possibility is that all 6 are even. Every pair a,b is either both 2(mod4) or both 0(mod4) since (2x)^2==0(mod4). If they are all 0(mod4), they can all be divided by 4 until a set is obtained which is all 2(mod4).

2(mod4)≡2 or 6 (mod8). If a and b are 2(mod8), a+b≡(4(mod8) and cannot be square. If a and b are 6(mod(8), a+b are 12(mod8) and cannot be square. In any set of 6 intergers ≡2(mod8), there will always be a pair both 2(mod8) or both 6(mod8).

Question: What is the largest set possible such that every 3 sum to a square? Are there any sets of 4 such that every 3 sum to a cube?

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