# Theorem in the Additive Number Theory

@inproceedings{ZivTheoremIT, title={Theorem in the Additive Number Theory}, author={Abraham Ziv} }

THEOREM. Each set of 2n-1 integers contains some subset of n elements the sum of which is a multiple of n. PROOF. Assume first n = p (p prime). Our theorem is trivial for p = 2, thus henceforth p > 2. We need the following LEMMA. Let p > 2 be a prime and A = {a,, a 2 ,. . ., a,} 2 5 s < p a s tegers each prime to p satisfying ca, a 2 (mod p). Then the set a i a + , s =10 or 1 contains at least s + 1 distinct congruence classes. +=1 We use induction. If s = 2, a, i a 2 , a, + a 2 are all…

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