On Partitions Of The Set Of Residue Classes With The Same Representation Functions

Let m be a positive integer and Zm={0,1,…m-1} be the set of residue classes modulo m.We define the ordering as 0<1<…<m-1,and a?b if and only if a=b or a<b.For A (?)Zm and n?Zm,Let R1(A,n)=|{(a,a')?A2:a+a'=n}|,R2(A,n)=|{(a,a')?A2:a+a'=n,a<a'}|,R3(A,n)=|{a,a')?A2:a+a'=n,a?a'|.Ri(A,n)(i=1,2,3)are called additive representation functions,Ri(i=1,2,3)in short.In 2020,Chen and Yan studied the partitions of the set of residue classes with the same representation function R2(A,n)in paper[On certain properties of partitions of Zm with the same representation function,Discrete Math.343(2020),111981].In this thesis,we study the partitions of the set of residue classes with the same representation functions R1(A,n)and R3(A,n).In chapter 2,we study the partitions of the set of residue classes with the same representation function R1(A,n)and obtain the following results:(1)Let m be a positive even integer.Let A,B(?) Zm with A U B=Zm and|A?B|=2 or m-2.Then RA(n)=RB(n)for all n ?Zm if and only if B=A+m/2.(2)Let m be a positive integer with 4|m.Then there exist two distinct sets A,B(?)Zm with A?B=Zm,B?A+m/2 and |A?B}=4 or m-4 such that RA(n)=RB(n)for all n?Zm.(3)Let m be a positive integer with m?2(mod 4).Let A,B(?)Zm with A?B=Zm and |A?B|=4 or m-4.Then RA(71)=RB(n)for all n?Zm if and only if B=A+m/2.In chapter 3,we study the partitions of the set of residue classes with the same representation function R3(A,n)and obtain the following results:(1)Let ??2 be an integer and m=2?.Let A,B(?)Zm with A?B=Zm and |A?B|=2.Then RA(n)=RB(n)for all n?Zm if and only if B=A+m/2.(2)Let Z4={0,1,2,3}.Let A={0,1,2} and B={0,1,3}.Then B?A+m/2 and RA(n)=RB(n)for all n? Zm.(3)Let p be an odd prime and m=2p.If 2 |ordp(2),then there exist two distinct sets A,B (?) Zm with A?B=Zm,|A?B|=2 and B?A+m/2 such that RA(n)=RB(n)for all n?Zm.(4)Let p be an odd prime and m=2p.Let A,B(?)Zm with A?B=Zm and |A?B|=2.If 2(?)ordp(2),then RA(n)=RB(n)for all n?Zm if and only if B=A+m/2.