TAILIEUCHUNG - Báo cáo toán học: "The Prime Power Conjecture is True for n

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Prime Power Conjecture is True for n | The Prime Power Conjecture is True for n 2 000 000 Daniel M. Gordon Center for Communications Research 4320 Westerra Court San Diego CA 92121 gordon@ Submitted August 11 1994 Accepted August 24 1994. Abstract The Prime Power Conjecture PPC states that abelian planar difference sets of order n exist only for n a prime power. Evans and Mann 2 verified this for cyclic difference sets for n 1600. In this paper we verify the PPC for n 2 000 000 using many necessary conditions on the group of multipliers. AMS Subject Classification. 05B10 1 Introduction Let G be a group of order v and D be a set of k elements of G. If the set of differences di dj contains every nonzero element of G exactly A times then D is called a v k A -difference set in G. The order of the difference set is n k A. We will be concerned with abelian planar difference sets those with G abelian and A 1. The Prime Power Conjecture PPC states that abelian planar difference sets of order n exist only for n a prime power. Evans and Mann 2 verified this for cyclic difference sets for n 1600. In this paper we use known necessary conditions for existence of difference sets to test the PPC up to two million. Section 2 describes the tests used and Section 3 gives details of the computations. All orders not the power of a prime were eliminated providing stronger evidence for the truth of the PPC. THE ELECTRONIC .JOURNAL OF COMBINATORICS 1 1994 R6 2 2 Necessary Conditions We begin by reviewing known necessary conditions for the existence of planar difference sets. The oldest is the Bruck-Ryser-Chowla Theorem which in the case we are interested in states Theorem 1 If n 1 2 mod 4 and the squarefree part of n is divisible by a prime p 3 mod 4 then no difference set of order n exists. A multiplier is an automorphism a of G which takes D to a translate g D of itself for some g 2 G. If a is of the form a x tx for t 2 z relatively prime to the order of G then a is called a numerical multiplier. Most .

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