# Computations on normal families of primes

Wong, Erick (1997) Computations on normal families of primes. Masters thesis, Simon Fraser University.

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We call a family of primes $P$ normal if it contains no two primes $p,q$ such that $p$ divides $q-1$. In this thesis we study two conjectures and their related variants. Guiga's conjecture is that \sum_{k=1}^{n-1} k^{n-1} \equiv n-1 \mod {n)$implies$n$is prime. We study a group of eight variants of this equation and derive necessary and sufficient conditions for which they hold. Lehmer's conjecture is that$\phi(n)|n-1$if and only if$n$is prime. This conjecture has been verified for up to 13 prime factors of$n$, and we extend this to 14 prime factors. We also examine the related condition$\phi(n)|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.