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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
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.
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
unknown title
, 2005
"... Abstract. We extend our previous computations to show that there are 585355 Carmichael numbers up to 1017. 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 C ..."
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Abstract. We extend our previous computations to show that there are 585355 Carmichael numbers up to 1017. 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.
RICHARD G.E. PINCH
, 1998
"... Abstract. We extend our previous computations to show that there are 246683 Carmichael numbers up to 10 16. 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 246683 Carmichael numbers up to 10 16. 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.
unknown title
"... 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.
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.
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
Notes by G.J.O. Jameson
"... Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is ..."
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Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is