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 cn=2n+3+(−1)n3. Then we prove that the numbers cn=2n+3+(−1)n3 are never prime when n is odd and n>1. Thus, the sequence simplifies to cm=2m+13 with m odd. Finally, we prove that if 2p+13 is prime, then p must be a prime.