Distinct Lengths Modular Zero-sum Subsequences: A Proof of Graham’s Conjecture
نویسندگان
چکیده
Let n be a positive integer and let S be a sequence of n integers in the interval [0, n − 1]. If there is an r such that any nonempty subsequence with sum ≡ 0 (mod n) has length = r, then S has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erdős and E. Szemerédi shows the validity of this conjecture if n is a large prime number.
منابع مشابه
On the existence of distinct lengths zero-sum subsequences
— In this note, we obtain a characterization of short normal sequences in a finite Abelian p-group, thus answering positively a conjecture of W. Gao for a variety of such groups. Our main result is deduced from a theorem of N. Alon, S. Friedland and G. Kalai, originally proved so as to study the existence of regular graphs in almost regular graphs. In the special case of elementary p-groups, Ga...
متن کاملOn the existence of zero-sum subsequences of distinct lengths
— In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao’s co...
متن کاملNote on a conjecture of Graham
An old conjecture of Graham stated that if [Formula: see text] is a prime and [Formula: see text] is a sequence of [Formula: see text] terms from the cyclic group [Formula: see text] such that all (nontrivial) zero-sum subsequences have the same length, then [Formula: see text] must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently ...
متن کاملPartial proof of Graham Higman's conjecture related to coset diagrams
Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...
متن کاملOn the Number of m-term Zero-Sum Subsequences∗
A sequence S of terms from an abelian group is zero-sum if the sum of the terms of S is zero. In 1961 Erdős, Ginzburg and Ziv proved that any sequence of 2m− 1 terms from an abelian group of order m contains an m-term zero-sum subsequence [10]. This sparked a flurry of generalizations, variations and extensions [1] [3] [7] [8] [11] [13] [14] [15] [16] [17] [18] [22] [26] [27] [28] [37]. Since a...
متن کامل