On certain properties of partitions of Zm with the same representation function.

For a given set S⊆Zm and n¯∈Zm, RS(n¯) is defined as the number of solutions of the equation n¯=s¯+s′¯ with unordered pair (s¯,s′¯)∈S2 and s¯≠s′¯. In this paper, we prove that if m is an even positive integer and not a power of 2 then there exist two distinct sets A,B with A∪B=Zm, |A∩B|=2, B≠A+m∕2¯ such that RA(n¯)=RB(n¯) for all n¯∈Zm. If m is a power of 2, A∪B=Zm, |A∩B|=2, then RA(n¯)=RB(n¯) for all n¯∈Zm if and only if B=A+m∕2¯.