Minimal size balanced sets mod p

A nonempty set S of residues modulo N is said to be balanced if for each x∈S, there is a d with 0<d≤N/2 such that x±dmodN both lie in S. We denote the minimum cardinality of a balanced set modulo N by α(N). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo p, when p is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence c_n=[Formula presented]. Then we prove that the numbers c_n=[Formula presented] are never prime when n is odd and n>1. Thus, the sequence simplifies to c_m=[Formula presented] with m odd. Finally, we prove that if [Formula presented] is prime, then p must be a prime.