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Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
The Pseudoprimes up to 10^13
, 1995
"... . There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a backtracking search for possible prime factorisatio ..."
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. There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a backtracking search for possible prime factorisations, and the computations checked by a sieving technique. 1 Introduction A (Fermat) pseudoprime (base 2) is a composite number N with the property that 2 N \Gamma1 j 1 mod N . For background on pseudoprimes and primality tests in general we refer to Bressoud [1], Brillhart et al [2], Koblitz [4], Ribenboim [12] and [13] or Riesel [14]. Previous tables of pseudoprimes were computed by Pomerance, Selfridge and Wagstaff [11]. We have shown that there are 38975 pseudoprimes up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 ; all have at most 9 prime factors. Let P (X) denote the number of pseudoprimes less than X and let P (d; X) denote the number with exactly d prime factors. In ...
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
The Carmichael Numbers up to 10^16
 Math. Comp
, 1993
"... 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 Carmich ..."
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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.
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
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.
RICHARD G.E. PINCH
, 2006
"... Abstract. We extend our previous computations to show that there are 1401644 Carmichael numbers up to 1018. 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 1401644 Carmichael numbers up to 1018. 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
, 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.