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Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
On Carmichael numbers in arithmetic progressions
 J. Aust. Math. Soc
"... We dedicate this paper to our friend Alf van der Poorten Assuming a weak version of a conjecture of HeathBrown on the least prime in a residue class, we show that for any coprime integers a and m> 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. 1 1 ..."
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We dedicate this paper to our friend Alf van der Poorten Assuming a weak version of a conjecture of HeathBrown on the least prime in a residue class, we show that for any coprime integers a and m> 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. 1 1
Constructing Carmichael numbers through improved subsetproduct algorihms
, 2012
"... Abstract. We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem tha ..."
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Abstract. We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the nonuniform distribution of primes p with the property that p − 1 divides a highly composite Λ. 1.
A new algorithm for constructing large Carmichael
 Ken Nakamula, Department of Mathematics and Information Sciences, Tokyo Metropolitan University, MinamiOsawa, Hachioji
, 1996
"... Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 11015 ..."
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Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors. 1.
The Carmichael numbers up to 10 21
"... We extend our previous computations to show that there are 20138200 Carmichael numbers up to 10 21. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmicha ..."
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We extend our previous computations to show that there are 20138200 Carmichael numbers up to 10 21. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael
ABSOLUTE QUADRATIC PSEUDOPRIMES
"... Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1. ..."
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Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.
The Carmichael numbers up to 10 20
"... We extend our previous computations to show that there are 8220777 Carmichael numbers up to 10 20. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichae ..."
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We extend our previous computations to show that there are 8220777 Carmichael numbers up to 10 20. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael
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"... Abstract. We extend our previous computations to show that there are 20138200 Carmichael numbers up to 1021. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of ..."
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Abstract. We extend our previous computations to show that there are 20138200 Carmichael numbers up to 1021. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael numbers. 1.
CARMICHAEL NUMBERS WITH SMALL INDEX (ABSTRACT)
"... A Carmichael number N is a composite number N with the property that for every b prime to N we have b N−1 ≡ 1 mod N. It follows that a Carmichael number N must be squarefree, with at least three prime factors, and that p − 1N − 1 for every prime p dividing N: conversely, any such N must be a Carmi ..."
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A Carmichael number N is a composite number N with the property that for every b prime to N we have b N−1 ≡ 1 mod N. It follows that a Carmichael number N must be squarefree, with at least three prime factors, and that p − 1N − 1 for every prime p dividing N: conversely, any such N must be a Carmichael number. For background on Carmichael numbers and details of previous computations we refer to our previous paper [3]: in that paper we described the computation of the Carmichael numbers up to 10 15 and presented some statistics. These computations have since been extended to 10 18 [4]. We define the Carmichael lambda function λ(N) to be the exponent of the multiplicative group (Z/N) ∗. The definition of Carmichael number is equivalent to the condition λ(N)N − 1. We define the index i(N) to be the integer (N − 1)/λ(N). Alford, Granville and Pomerance [1] have shown that there are infinitely many Carmichael numbers, but their argument produces numbers N with i(N) ∼ N 1−ɛ. We consider how small the index of a Carmichael number can be. Somer [5] proved a result implying that i(N) → ∞ as N → ∞. In this paper