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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.
On a variation of a congruence of Subbarao
, 2010
"... To the memory of Alf van der Poorten Here, we study positive integers n such that nφ(n) ≡ 2 (mod σ(n)), where φ(n) and σ(n) are the Euler function and the sum of divisors function of the positive integer n, respectively. We give a general ineffective result showing that there are only finitely many ..."
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To the memory of Alf van der Poorten Here, we study positive integers n such that nφ(n) ≡ 2 (mod σ(n)), where φ(n) and σ(n) are the Euler function and the sum of divisors function of the positive integer n, respectively. We give a general ineffective result showing that there are only finitely many such n whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes 2 and 3 we use continued fractions to find all such positive integers n. 1 1
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.
PSEUDOPRIMES, PERFECT NUMBERS, AND A PROBLEM OF LEHMER
, 1996
"... Two classical problems In elementary number theory appear, at first, to be unrelated. The first, posed by D. H. Lehmer in [7], asks whether there is a composite integer N such that {N) divides Nl, where &(N) is Euler's totient function. This question has received considerable attention and it ha ..."
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Two classical problems In elementary number theory appear, at first, to be unrelated. The first, posed by D. H. Lehmer in [7], asks whether there is a composite integer N such that </>{N) divides Nl, where &(N) is Euler's totient function. This question has received considerable attention and it has been demonstrated that such an integer, if it exists, must be extraordinary.
A NOTE CONCERNING THOSE n FOR WHICH 4>(n) + 1 DIVIDES n
, 1987
"... In [3, p. 52], Richard Guy gives the following problem of Schinzel: If p is an odd prime and n = 2 or p or 2p, then (cj)(n) + 1) \n, where is Euler's totient function. Is this true for any other n? We shall show that this question is closely related to a much older problem due to Lehmer [4]: whe ..."
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In [3, p. 52], Richard Guy gives the following problem of Schinzel: If p is an odd prime and n = 2 or p or 2p, then (cj)(n) + 1) \n, where <J> is Euler's totient function. Is this true for any other n? We shall show that this question is closely related to a much older problem due to Lehmer [4]: whether or not there exist composite n such that (J) (n)  (n 1). It will turn out that if there are no such composite n, then Schinzel! s are the only solutions of his problem; if there are other solutions of Schinzel? s problem, then they have at least 15 distinct prime factors. Let oo(n) denote the number of distinct prime factors of n. More specifically, we shall prove the following. Theorem: Let n be a natural number and suppose (<J)(n) + l)n. Then one of the following is true. (i) n = 2 or p or 2p, where p is an odd prime. (ii) n = mt, where m = 3, 4, or 6, gcd(/?7, t) = 1, and t 1 = 2$(t) [so that co(t)> 14]. (iii) n = mt, where gcd(m5 t) = 1, <$>(m) = j> 4, and t 1 = j<$>(t) [so that oo(t)> 140]. Proof: Since ($(ri) + 1)\n9 we have m(($)(n) + 1) = n (1) for some natural number 777. Let t = cj>(ft) + 1 an d ^ = gcd(/?7, t). Then, using (1) and an easy and wellknown result (Apostol [1, p. 28]), (>(n) = <\>(mt) = \ / ^ \ ( 2) Since d\m> we have §(d) \$ ( m) s o that $(m)/$(d) is an integer. Then, from (2), d\$(n); but, by definition, d\($(n) + 1). Hence d = 1. Thus, we have n = mt, where t = <()(n) + 1 = M/??t) + 1 = <j>(m)<K£) + 1.
INEQUALITIES RELATED TO THE UNITARY ANALOGUE OF LEHMER PROBLEM
, 2006
"... ABSTRACT. Observing that φ(n) divides n − 1 if n is a prime, where φ(n) is the well known Euler function, Lehmer has asked whether there is any composite number n with this property. For this unsolved problem, partial answers were given by several researchers. Considering the unitary analogue φ ∗ (n ..."
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ABSTRACT. Observing that φ(n) divides n − 1 if n is a prime, where φ(n) is the well known Euler function, Lehmer has asked whether there is any composite number n with this property. For this unsolved problem, partial answers were given by several researchers. Considering the unitary analogue φ ∗ (n) of φ(n), Subbarao noted that φ ∗ (n) divides n − 1, if n is the power of a prime; and sought for integers n other than prime powers which satisfy this condition. In this paper we improve two inequalities, established by Subbarao and Siva Rama Prasad [5], to be satisfied by n for φ ∗ (n) which divides n − 1. [5] M.V. Subbarao and V. Siva Rama Prasad, Some analogues of a Lehmer problem on the totient