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Topic: Set of 2n1 integers (Read 6217 times) 

Eigenray
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Set of 2n1 integers
« on: Dec 23^{rd}, 2005, 10:39am » 
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Show that any set of 2n1 integers always has a subset of size n, the sum of whose elements is divisible by n.


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SMQ
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Re: Set of 2n1 integers
« Reply #1 on: Dec 23^{rd}, 2005, 2:03pm » 
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Well, I don't know how much this actually helps, but here's an inductive proof for composite n: Let k, 1 < k < n be a divisor of n. Choose any 2n/k  1 integers from the set. There exists subset of n/k integers whose sum is a multiple of n/k. Remove those n/k integers from consideration and pick any other 2n/k  1 integers. Continuing like this it is possible to pick 2k  1 nonoverlapping sets of integers each of which sums to a multiple of n/k. But, considering the sums of those 2k  1 sets, there exists a set of k sets whose sum is a multiple of k. The set of all n integers in those k sets of n/k integers is a multiple of both n/k and k, and therefore a multiple of k*n/k = n. SMQ


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Icarus
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Re: Set of 2n1 integers
« Reply #2 on: Dec 23^{rd}, 2005, 5:13pm » 
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on Dec 23^{rd}, 2005, 2:03pm, SMQ wrote:... is a multiple of both n/k and k, and therefore a multiple of k*n/k = n. 
 if k is relatively prime to n. The proof is still good except for powers of primes, provided it can be shown that the powers of primes also work.


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SMQ
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Re: Set of 2n1 integers
« Reply #3 on: Dec 23^{rd}, 2005, 8:05pm » 
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OK, after choosing nonoverlapping subsets, rather than considering the set of the sums of the subsets, consider the set of the sums of the subsets each divided by n/k. Clearly this is still a set of 2k  1 integers and so must itself contain at least one subset of size k whose sum is a multiple of k. Thus, even for prime powers, the set consisting of all members of those k subsets will contain independent factors of n/k and k, and therefore be a multiple of n. But without a proof for prime n, there's still no base case for the induction for composite n... SMQ


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Barukh
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Re: Set of 2n1 integers
« Reply #4 on: Dec 24^{th}, 2005, 5:37am » 
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The proof for the prime n may be as follows. Assume to the contrary that for k = 1, …, C(2n1, n), S_{k} > 0, where S_{k} is the sum of kth subset of n numbers, modulo n. Then, by Fermat’s theorem, S_{k}^{n1} = 1. Therefore, S_{k}^{n1} = C(2n1, n). The r.h.s. equals 1 modulo n (why?). In the l.h.s. every of 2n1 numbers appears C(2n2, n1) times, which is 0 modulo n (why?). A contradiction.

« Last Edit: Dec 25^{th}, 2005, 8:31pm by Icarus » 
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Aryabhatta
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Re: Set of 2n1 integers
« Reply #5 on: Dec 24^{th}, 2005, 10:25am » 
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I have seen a proof (I think due to Noga Alon) for the prime number case which uses the ChevalleyWarning Theorem

« Last Edit: Dec 24^{th}, 2005, 10:25am by Aryabhatta » 
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Barukh
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Re: Set of 2n1 integers
« Reply #6 on: Dec 25^{th}, 2005, 4:50am » 
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I think the following side problem is relevant here: Given natural numbers n, m and a prime p, let s = floor(n/p), t = floor(m/p). Prove that C(n, m) = C(s, t)C(n mod p, m mod p) mod p.


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ecoist
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Re: Set of 2n1 integers
« Reply #7 on: Feb 28^{th}, 2006, 3:22pm » 
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How does this problem differ from the pigeonhole problem: Given any set of n integers, there exists a subset whose elements sum to a multiple of n. The comments I read suggest that the 2n1 problem is harder than the above problem, which has a short and sweet solution. Am I missing something?


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Icarus
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Re: Set of 2n1 integers
« Reply #8 on: Feb 28^{th}, 2006, 3:42pm » 
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Yes  that the set has to have exactly n elements. For example, consider your result applied to the following sets {1, 2}, {1, 3}. The subset adding to an even number for the first is {2}, while the subset adding to an even number for the second is {1, 3}. In Eigenray's problem, all the subsets must have exactly n elements.


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Eigenray
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Re: Set of 2n1 integers
« Reply #9 on: Feb 22^{nd}, 2008, 9:50am » 
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on Dec 24^{th}, 2005, 5:37am, Barukh wrote: S_{k}^{n1} = C(2n1, n). The r.h.s. equals 1 modulo n (why?). In the l.h.s. every of 2n1 numbers appears C(2n2, n1) times, which is 0 modulo n (why?). 
 Hmm... did I think this was obvious 2 years ago? Anyway, we can show by induction that _{S} (S)^{r} = 0 mod p, where the outer sum is over all ksubsets S of (p+k1) elements, and r < k, for k=1,...,p. We can also use the version of Chevalley Warning for simultaneous solutions here. Hint: We have to make 2p1 independent choices that satisfy 2 conditions.

« Last Edit: Feb 22^{nd}, 2008, 9:51am by Eigenray » 
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Eigenray
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Re: Set of 2n1 integers
« Reply #10 on: Feb 22^{nd}, 2008, 3:38pm » 
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I think Barukh's argument generalizes to show: Suppose pk, C(n, k) is not divisible by p, and p  (n+1)C(np, kp). Then for any n integers, there is a subset of size exactly k whose sum is divisible by p. The idea is that p  C(nt, kt) for 0<t<p, but not for t=0. Actually, the above phrasing is stupid, because if the result holds for some n it obviously holds for any larger n. Therefore TFAE: A) Every set of n integers has a subset of size k whose sum is divisible by p B) p  k and n k+p1.

« Last Edit: Feb 22^{nd}, 2008, 3:45pm by Eigenray » 
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