循环差集存在的一个必要条件和素数的一个性质

定义　设υ,ｋ,λ是正整数.模υ的ｋ个互不同余的整数组成的集合Ｄ={ｄ1,ｄ2,…,ｄｋ}叫做一个(υ,ｋ,λ)-循环差集,如果对于每一个α0(ｍｏｄυ),恰好在Ｄ中有λ个有序对(ｄｉ,ｄｊ),使得α≡ｄｉ-ｄｊ(ｍｏｄυ).由于一个循环差集可以展开为一个循环对称区组设计,由著名的ＢｒｕｃｋＲｙｓｅｒＣｈｏｗｌａ定理,有如下结论:定理11]　设1≤λ<ｋ<υ-1.若(υ,ｋ,λ)-差集存在,则ⅰ)λ(υ-1)=ｋ(ｋ-1),ⅱ)当υ为偶数时,ｋ-λ为平方数;当υ为奇数时,不定方程ｚ2=(ｋ-λ)ｘ2 (-1)(υ-1)/2λｙ2(1)有不全为零的整数解ｘ,ｙ,ｚ.判定不定方程(1)…

A Necessary Condition for the Existence of Cyclic Difference Set and a Property of Prime Number

In this paper，we prove that when υ≡0（mod 3），the necessary condition for the existence of cyclic（υ，k，λ）－difference set is that the equation n＝x2＋y2＋xy has a nonnegative solution in integers x，y（where n＝k－λ）．From this conclusion we obtain that i）there do not exist cyclic（υ，k，λ）－difference set when k－λ≡6 or 10（mod 12）；ii）when p≡1（mod 3）is a prime number，then the equation p＝x2＋y2＋xy has a nonnegative solution in integers x，y．