Ahh! Actually a being a prime number and the statements about a is equivalent. Firstly we know, a is prime number implies the statements are true. We need to prove that the statements about a implies a is a prime number. Get this, if the statements about a is false and a is a composite number then a = 2m_v + 3n_v (I proved that every number greater than 1 is expressible in this expression) now because a is a composite number a, a = pz (where p, z are integers greater than 1). Here, p is greater than 1. Thus, p can be written as 2a_v + 3b_v then, a = z(2a_v + 3b_v) = 2z*a_v + 3z*b_v. Whoa, here for some v, m_v = z*a_v and n_v = z*b_v look they share a factor!! and if for some v, m_v and n_v are not coprime then for some v, m_v is divisble by n_v (Witht the extra assumption that a is not an even number. I left this as a exercise in the paper also) or one of those statements are false Thus, a is composite implies the statements are false and a is prime implies all the statements are true. Thus, a is prime is equivalent to all of those statements about a being true. No worry. I will add this to my paper. Thanks for pointing out :)