Computations on Normal Families of Primes

We call a family of primes P normal if it contains no two primes p; q such that p divides q \Gamma 1. In this thesis we study two conjectures and their related variants. Giuga's conjecture is that P n\Gamma1 k=1 k n\Gamma1 j n \Gamma 1 (mod n) implies n is prime. We study a group of eight variants of this equation and derive necessary and sufficient conditions for which they hold. Lehmer's conjecture is that OE(n) j n \Gamma 1 if and only if n is prime. This conjecture has been verified for up to 13 prime factors of n, and we extend this to 14 prime factors. We also examine the related condition OE(n) j n + 1 which is known to have solutions with up to 6 prime factors and extend the search to 7 prime factors. For both of these conjectures the set of prime factors of any counterexample n is a normal family, and we exploit this property in our computations