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## #1 2011-07-26 19:22:36

Member
Registered: 2011-07-23
Posts: 16

### no sets of six integers with every pair summing to a square

Supposing that there is a set (a,b,c,d,e,f) and a+b, a+c, a+d, b+c... are all squares. Then each member of the set can be paired with 5 others but a+b=b+a, so there are 5*6/2=15 combinations. 5(a+b+c+d+e+f)=sum of all 15 squares.
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?

Last edited by namealreadychosen (2011-07-26 19:29:10)

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## #2 2011-07-26 21:12:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: no sets of six integers with every pair summing to a square

Hi;

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.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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