There are 105212 Carmichael numbers up to 1015: we describe the calculations. The numbers were generated by a back-tracking search for possible prime factorisations, and the computations checked by searching selected ranges of integers directly using a sieving technique, together with a ``large prime variation''.
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Zbl. 780.11069
Math Review 93m:11137
Arxiv:math.NT/9803082 or here.
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See the full paper in DVI or PS format.
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See a poster for the ANTS VII meeting in PS format.
We further extend our computations to show that there are 20138200 Carmichael numbers up to 1021.
See a poster for the ANTS VIII meeting in PDF format.
See the the full paper in PDF format.
carmichael-16.gz
All Carmichael numbers up to 1016 in GZIP format
carmichael17.gz
All Carmichael numbers between 1016 and 1017 in GZIP format
carmichael18.gz
All Carmichael numbers between 1017 and 1018 in GZIP format
car3-18.gz
All Carmichael numbers with 3 prime factors up to 1018 in GZIP format
Proceedings 6th IMA Conference on Coding and Cryptography, Cirencester 1997, (ed. M. Darnell) Springer Lecture Notes in Computer Science 1355 (1997) 265--269
We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected.
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4th International Algorithmic Number Theory Symposium, ANTS-IV, Leiden, The Netherlands, 2--7 July 2000; Springer Lecture Notes in Computer Science 1838 (2000), 459-474
There are 38975 Fermat pseudoprimes (base 2) up to 1011, 101629 up to 1012 and 264239 up to 1013: we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique..
See the full paper in PDF format. (Copyright © Springer-Verlag 2000)
psp-12.gz
The 264239 pseudoprimes up to 1013 in GZIP format
even-12
The 155 even pseudoprimes
up to 1012.
Some preprints are available in the Number Theory section of the eprint Arxiv: try here or use the UK mirror site at Southampton
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