## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  349.10046
Autor:  Erdös, Paul; Szemeredi, E.
Title:  On a problem of Graham. (In English)
Source:  Publ. Math., Debrecen 23, 123-127 (1976).
Review:  The following conjecture is considered. Let p be a prime, and let a1, ... ,ap be non-zero residues (mod p) such that, if sum \epsiloniai (\epsiloni = 0 or 1, not all \epsiloni = 0) is a multiple of p, then sum \epsiloni is uniquely determined. Then there are at most two distinct residues among the ai. A proof of this conjecture, for sufficiently large p, is presented. It is remarked by the authors that the proof is surprisingly complicated. The lack of clarity in the exposition in no way helps the reader to overcome this difficulty. A preliminary theorem is proved, which `easily implies Graham's conjecture in case each residue occurs with a multiplicity < \eta0p'. What this means is that the theorem implies that under the assumptions of Graham's conjecture, some residue must occur at least \eta0p times. This is the first step in the proof of the conjecture. The remaining steps are difficult, using theorems of Dirichlet, Cauchy-Davenport and Erdös-Heilbronn. Finally, the reader is left wondering if Graham is R. L. Graham, and even if the first author is P. or E. Erdös.
Reviewer:  I.Anderson