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Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dicks ..."
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
Computations on Normal Families of Primes
, 1997
"... 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 varian ..."
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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 \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.