Discover the Proof for Primes: Solving the Mystery of Interesting Sets"

[newchie]

Would like to see a proof for the following question.

Let p be a prime number. Define a set interesting if it has p+2 (not necessarily distinct) positive integers such than the sum of any p numbers is a multiple of each of the other two. Find all interesting sets.

 

[mathwonk]

how interesting is this?

 

[dodo]

how interesting is this?

))

If one such collection is 'interesting', such as {1,1,1,1,3} for p=3, then any scaled-up version is also 'interesting': {2,2,2,2,6}, {10,10,10,10,30}, or the like. So call these collections 'primitive' (there goes another wordo) if they don't have any common factor among all numbers (the numbers could still be non-coprime when taken pairwise).

The issue is that, if you try with a computer, the only 'primitive' collections appear to be: A) the "all ones" collection, (p+2 ones), or B) the "almost all ones" (p+1 ones and one 'p'). For example, for p=3, you find only {1,1,1,1,1} and {1,1,1,1,3} (assume the order is irrelevant, otherwise place the '3' in any of the 5 possible positions).

It's expensive to try with the computer for any but small numbers, so the issue is: is this the only solution, of is any other possible for larger numbers?

 

[Norwegian]

The only interesting primitive sets are {1,1,1,...,1} and {p,1,1,...,1}.

Let {x₁,...,x_p+2} be interesting and primitive, and let S_k=(Ʃx_i) - x_k.

First observe that if one number, say x_k, is divisible by p, then all other x_i are congruent S_k modulo p. (Since x_k|S_k-x_i.)

Conclude that at most one of the x_i are divisible by p.

For i≠j, we have S_j-x_i=a_ix_j, and this equation summed over all i (with i≠j), gives pS_j=ax_j.
In particular, since x_j|pS_j and x_j|p(S_j-x_i), we must have x_j|px_i for all i and j.

Conclude that all x_i not divisible by p must be equal, and if any x_j is divisble by p, it must be p times as much.

 

[dodo]

There should be an :applauding: smilie here. It took me some time, but I think I followed the whole thing. Nice!

 

[epsi00]

There should be an :applauding: smilie here. It took me some time, but I think I followed the whole thing. Nice!

I second that.