## Building Pseudoprimes With A Large Number Of Prime Factors (1995)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Guillaume95buildingpseudoprimes,

author = {D. Guillaume and F. Morain},

title = {Building Pseudoprimes With A Large Number Of Prime Factors},

year = {1995}

}

### OpenURL

### Abstract

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.

### Citations

72 | There are infinitely many Carmichael numbers
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...ary, 1992 (this formed [14]). At this point, we learned that, spurred by the work of Zhang [42], Alford, Granville and Pomerance had just proved that there is an infinite number of Carmichael numbers =-=[1]-=-. Then, around May 15, 1992, we heard about the work of Keller and Loh and Niebuhr. (We note that very few people were aware of this work.) Though part of our work is subsumed by that of Loh and Niebu... |

70 |
Prime numbers and computer methods for factorization
- Riesel
- 1985
(Show Context)
Citation Context ... number C = p 1 \Delta \Delta \Delta p r with rs3 such that C \Gamma 1 j 0 mod (p i \Gamma 1) for 1sisr: (1) An alternative statement is that (C) j C \Gamma 1 wheresdenotes Carmichael's function (see =-=[36]-=-). Recall that (p e ) = '(p e ) for p an odd prime or p = 2 and es2 (' is Euler's totient function), (2 e ) = 2 e\Gamma2 for e ? 2 and / r Y i=1 p e i i ! = lcm((p e 1 1 ); : : : ; (p er r )): From th... |

44 |
An extended theory of Lucas’ functions
- Lehmer
- 1930
(Show Context)
Citation Context ...roots of the equation X 2 \Gamma PX + Q = 0. Then, the Lucas sequences are defined as V n (P; Q) = ff n + fi n ; U n (P; Q) = ff n \Gamma fi n ff \Gamma fi : These sequences have many properties (see =-=[21]-=- or [35]), including arithmetical ones. Let ffl(N ) denote the Jacobi symbol / \Delta N ! . Theorem 4.1 Let p be an odd prime such that p - Q\Delta. Then U p\Gammaffl(p) (P; Q) j 0 mod p: A Lucas pseu... |

33 |
Highly composite numbers
- Ramanujan
- 2000
(Show Context)
Citation Context ...e exponents A that yield highly composite values ofs(recall that a highly composite number n is such that m ! n implies that m has fewer divisors than n; those numbers were first studied by Ramanujan =-=[34]-=-; see also [29]). 3.4 Remarks 3.4.1 Using outside primes While working out our ideas, we tried to modify the algorithm so as to produce Carmichael numbers all of whose prime factors where in S () but ... |

31 |
The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group
- Dickson
(Show Context)
Citation Context ... 18026-digit number with 2461 prime factors. 4.3 Strong Dickson pseudoprimes 4.3.1 Definition and properties Let c be an integer. The Dickson polynomial of parameter c and degree n is defined as (see =-=[8, 20, 23]-=-) g n (x; c) = bn=2c X i=0 n n \Gamma i / n \Gamma i i ! (\Gammac) i x n\Gamma2i : An odd composite integer N is called a strong Dickson pseudoprime (sDpp(c) in short) if for all m in Z: g N (m; c) j ... |

29 |
The book of prime number
- Ribenboim
- 1989
(Show Context)
Citation Context ...ests, where they can be seen as worst cases for the Fermat compositeness test (see [32]). We denote by C(x) the number of Carmichael numbers up to x. Many properties of these numbers are described in =-=[35]-=-. Recent tables of Carmichael numbers include that of Keller up to 10 13 (see [17]), that of Jaeschke [16] up to 10 12 (Jaeschke gave C(10 12 ) = 8238 but the correct value was found by Keller: C(10 1... |

24 |
Carmichael’s lambda function
- Erdös, Pomerance, et al.
- 1991
(Show Context)
Citation Context ... that can solve more difficult problems. Acknowledgments. We thank J.-L. Nicolas for his invaluable help and support while working out the results and writing the paper; G. Tenenbaum for pointing out =-=[10]-=- and A. Schinzel for his great culture on Davenport's problem; L. Reboul who brought Giuga's problem to our attention; R. Lercier who helped factoring some of the numbers involved; R. Pinch for making... |

17 | Note on a new number theory function - Carmichael - 1909 |

17 |
Algebra of polynomials
- Lausch, Nöbauer
- 1973
(Show Context)
Citation Context ... 18026-digit number with 2461 prime factors. 4.3 Strong Dickson pseudoprimes 4.3.1 Definition and properties Let c be an integer. The Dickson polynomial of parameter c and degree n is defined as (see =-=[8, 20, 23]-=-) g n (x; c) = bn=2c X i=0 n n \Gamma i / n \Gamma i i ! (\Gammac) i x n\Gamma2i : An odd composite integer N is called a strong Dickson pseudoprime (sDpp(c) in short) if for all m in Z: g N (m; c) j ... |

13 | On composite numbers P which satisfy the Fermat congruence a P −1 ≡ 1 mod P - Carmichael - 1912 |

11 |
On Fermat T s Simple Theorem
- Chernick
(Show Context)
Citation Context ...und Carmichael numbers with 12 and up to 18 prime factors. Pinch found the smallest Carmichael numbers with up to 20 prime factors [30]. Wagstaff [37] used the so-called "universal forms" of=-= Chernick [6]-=- to find large Carmichael numbers. Woods and Huenemann [39] gave larger numbers using the extension Theorem. Dubner [9] found even larger Carmichael numbers using a modification of the universal forms... |

11 |
Dickson polynomials
- Lidl, Mullen, et al.
- 1993
(Show Context)
Citation Context ... 18026-digit number with 2461 prime factors. 4.3 Strong Dickson pseudoprimes 4.3.1 Definition and properties Let c be an integer. The Dickson polynomial of parameter c and degree n is defined as (see =-=[8, 20, 23]-=-) g n (x; c) = bn=2c X i=0 n n \Gamma i / n \Gamma i i ! (\Gammac) i x n\Gamma2i : An odd composite integer N is called a strong Dickson pseudoprime (sDpp(c) in short) if for all m in Z: g N (m; c) j ... |

10 |
New primality criteria and factorizations of 2 m \Sigma 1
- Brillhart, Lehmer, et al.
- 1975
(Show Context)
Citation Context ...nd p \Gamma 1 = s Y i=1 q ff i i ; 0sff isA i ) : An advantage of this construction is that proving the primality of such numbers p is quite simple since the factorization of p \Gamma 1 is known (see =-=[3]-=-). Loh decided to choose the values of the exponents A that yield highly composite values ofs(recall that a highly composite number n is such that m ! n implies that m has fewer divisors than n; those... |

9 | The distribution of Lucas and elliptic pseudoprimes
- Gordon, Pomerance
- 1991
(Show Context)
Citation Context ... j \Gamma1 mod , but ffl(K) = \Gammaffl(P (S 43 )) Q i ffl(` i ), so K is not a 43-Lpsp. 4.2 Elliptic pseudoprimes 4.2.1 Theory For the definition of elliptic pseudoprimes, we refer to [12] (see also =-=[13]-=-). For our purpose, it is enough to cite the following result. Let D be an integer among f3; 4; 7; 8; 11; 19; 43; 67; 163g. Proposition 4.1 Let N be a squarefree composite number such that / \GammaD N... |

9 |
The pseudoprimes to 25 · 10
- Pomerance, Selfridge, et al.
- 1980
(Show Context)
Citation Context ...amma1 j 1 mod C holds for all values of a prime to C. These numbers are of interest in the study of pseudoprimality tests, where they can be seen as worst cases for the Fermat compositeness test (see =-=[32]-=-). We denote by C(x) the number of Carmichael numbers up to x. Many properties of these numbers are described in [35]. Recent tables of Carmichael numbers include that of Keller up to 10 13 (see [17])... |

9 |
A Probabilistic Primality Test Based on the Properties of Certain Generalized Lucas Numbers
- Porto, Filipponi
(Show Context)
Citation Context ...doprime (sDpp(c) in short) if for all m in Z: g N (m; c) j m mod N: (4) A strong Fibonacci pseudoprime (sF-psp in short) is simply an sDpp(\Gamma1): these numbers were first introduced and studied in =-=[33]-=-. Extending a result of [25], Kowol proved the following [19] Theorem 4.3 Let N = p 1 p 2 : : : p r be the prime factorization of an odd, squarefree number N prime to c and e i denote the order of c m... |

8 |
Some Remarks on Strong Fibonacci Pseudoprimes." Applicable Algebra in Eng
- Lidl, Muller, et al.
(Show Context)
Citation Context ...f for all m in Z: g N (m; c) j m mod N: (4) A strong Fibonacci pseudoprime (sF-psp in short) is simply an sDpp(\Gamma1): these numbers were first introduced and studied in [33]. Extending a result of =-=[25]-=-, Kowol proved the following [19] Theorem 4.3 Let N = p 1 p 2 : : : p r be the prime factorization of an odd, squarefree number N prime to c and e i denote the order of c modulo p i . Then N is a sDpp... |

7 |
On Euler’s totient function
- LEHMER
- 1932
(Show Context)
Citation Context ...first natural" discriminant. For this, we would need the same construction as above, but this time we insist on l 2 (p) js\Gamma . Even this seems out of reach for our method. 5.2 Lehmer's proble=-=m In [22]-=- Lehmer raised the following question: does there exist a composite integer N for which '(N) j N \Gamma 1? Such a number must be a Carmichael number. In view of [26, 18, 7] (see also [35]), such a num... |

5 |
On the number of elliptic pseudoprimes
- Gordon
- 1989
(Show Context)
Citation Context ...01) satisfies K j \Gamma1 mod , but ffl(K) = \Gammaffl(P (S 43 )) Q i ffl(` i ), so K is not a 43-Lpsp. 4.2 Elliptic pseudoprimes 4.2.1 Theory For the definition of elliptic pseudoprimes, we refer to =-=[12]-=- (see also [13]). For our purpose, it is enough to cite the following result. Let D be an integer among f3; 4; 7; 8; 11; 19; 43; 67; 163g. Proposition 4.1 Let N be a squarefree composite number such t... |

4 |
Building Carmichael numbers with a large number of prime factors and generalization to other numbers
- Guillaume, Morain
- 1992
(Show Context)
Citation Context ... the end of 1991, we had built Carmichael numbers with up to 5000 prime factors, as well as other numbers that will be described later. The paper was finished at the end of January, 1992 (this formed =-=[14]-=-). At this point, we learned that, spurred by the work of Zhang [42], Alford, Granville and Pomerance had just proved that there is an infinite number of Carmichael numbers [1]. Then, around May 15, 1... |

4 |
The Number of Distinct Prime Factors for Which oe(N
- Kishore
- 1977
(Show Context)
Citation Context ...ur method. 5.2 Lehmer's problem In [22] Lehmer raised the following question: does there exist a composite integer N for which '(N) j N \Gamma 1? Such a number must be a Carmichael number. In view of =-=[26, 18, 7]-=- (see also [35]), such a number N must have a large number of prime factors (at least 14 and many more if 3 j N ). It is tempting to look at our numbers and test whether they satisfy the condition. Ho... |

4 |
On Strong Dickson Pseudoprimes
- Kowol
- 1992
(Show Context)
Citation Context ...mod N: (4) A strong Fibonacci pseudoprime (sF-psp in short) is simply an sDpp(\Gamma1): these numbers were first introduced and studied in [33]. Extending a result of [25], Kowol proved the following =-=[19]-=- Theorem 4.3 Let N = p 1 p 2 : : : p r be the prime factorization of an odd, squarefree number N prime to c and e i denote the order of c modulo p i . Then N is a sDpp(c) if and only if for all i, one... |

4 |
Müller: Primality testing with Lucas functions
- Lidl, B
- 1993
(Show Context)
Citation Context ...sF-psp N is a Carmichael number for which 2(p i + 1) j N \Gamma 1 (type I) or 2(p i + 1) j N \Gamma p i for all i (type II). Other characterisations of sDpp(c) are given in [23, Chapter 7]. Following =-=[24]-=-, if N is a sDpp(c) for all c such that gcd(c; N) = 1, then N is called superstrong Dickson pseudoprime (ssDpp in short). The authors prove: Theorem 4.4 An odd composite N is a ssDpp if and only if (i... |

4 | The pseudoprimes up to 10 - Pinch - 1992 |

4 |
Numerical computation of Carmichael numbers
- Yorinaga
- 1978
(Show Context)
Citation Context ... = 8241; this is in agreement with our own calculations and that of Pinch) and by Pinch up to 10 15 (see [30]) and more recently up to 10 16 (email to the NMBRTHRY mailing list, April 1993). Yorinaga =-=[40, 41] has found-=- many Carmichael numbers using several methods including Chernick 's "extension theorem" (see Section 2). In particular, he found Carmichael numbers with 12 and up to 18 prime factors. Pinch... |

3 |
The Carmichael numbers to 10
- Jaeschke
- 1990
(Show Context)
Citation Context ...) the number of Carmichael numbers up to x. Many properties of these numbers are described in [35]. Recent tables of Carmichael numbers include that of Keller up to 10 13 (see [17]), that of Jaeschke =-=[16]-=- up to 10 12 (Jaeschke gave C(10 12 ) = 8238 but the correct value was found by Keller: C(10 12 ) = 8241; this is in agreement with our own calculations and that of Pinch) and by Pinch up to 10 15 (se... |

3 |
The Carmichael numbers to 10
- Keller
- 1988
(Show Context)
Citation Context ... [32]). We denote by C(x) the number of Carmichael numbers up to x. Many properties of these numbers are described in [35]. Recent tables of Carmichael numbers include that of Keller up to 10 13 (see =-=[17]-=-), that of Jaeschke [16] up to 10 12 (Jaeschke gave C(10 12 ) = 8238 but the correct value was found by Keller: C(10 12 ) = 8241; this is in agreement with our own calculations and that of Pinch) and ... |

3 |
On Numbers Analogous to the Carmichael Numbers
- Williams
- 1977
(Show Context)
Citation Context ...a. Then U p\Gammaffl(p) (P; Q) j 0 mod p: A Lucas pseudoprime for parameters (P; Q) is an odd composite integer N which satisfies U N \Gammaffl(N ) j 0 mod N . Let \Delta be a fixed integer. Williams =-=[38]-=- studied the properties of the numbers N for which U N \Gammaffl(N ) j 0 mod N for all choices of integers (P; Q) such that (P; Q) = 1, P 2 \Gamma 4Q = \Delta and (N; \DeltaQ) = 1. Let us call such a ... |

2 |
Carmichael numbers with a large number of prime factors
- oh
- 1988
(Show Context)
Citation Context ... D'el'egation G'en'erale pour l'Armement. Keywords: Carmichael numbers, pseudoprimes, Lucas pseudoprimes, elliptic pseudoprimes, strong Dickson pseudoprimes. mailing list, July 1994). Loh and Niebuhr =-=[27, 28]-=- have built numbers with up to 1; 101; 518 factors (this particular result was communicated to us by Keller). Recently, Zhang [42] has built a Carmichael number with 1305 prime factors and 8340 digits... |

2 |
The Carmichael numbers to 10
- Pinch
- 1993
(Show Context)
Citation Context ...p to 10 12 (Jaeschke gave C(10 12 ) = 8238 but the correct value was found by Keller: C(10 12 ) = 8241; this is in agreement with our own calculations and that of Pinch) and by Pinch up to 10 15 (see =-=[30]) and more-=- recently up to 10 16 (email to the NMBRTHRY mailing list, April 1993). Yorinaga [40, 41] has found many Carmichael numbers using several methods including Chernick 's "extension theorem" (s... |

1 |
Note on a conjecture on prime numbers
- Bedocchi
- 1985
(Show Context)
Citation Context ...t N j 1 N \Gamma1 + 2 N \Gamma1 + \Delta \Delta \Delta + (N \Gamma 1) N \Gamma1 + 1: Equivalently, N must be a Carmichael number and satisfy p j N ) N \Gamma 1 p \Gamma 1 j 1 mod p 2 : (See [35]). By =-=[2]-=-, such an N must be greater than 10 1700 . We did not find any Carmichael number built with our method that satisfies this property. 6 Conclusions Let us come back to the idea of the algorithm. We can... |

1 |
On the number of prime factors of n if OE(n) j (n \Gamma 1). Nieuw Archief voor Wiskunde (3) XXVIII
- Cohen, Hagis
- 1980
(Show Context)
Citation Context ...ur method. 5.2 Lehmer's problem In [22] Lehmer raised the following question: does there exist a composite integer N for which '(N) j N \Gamma 1? Such a number must be a Carmichael number. In view of =-=[26, 18, 7]-=- (see also [35]), such a number N must have a large number of prime factors (at least 14 and many more if 3 j N ). It is tempting to look at our numbers and test whether they satisfy the condition. Ho... |

1 |
A new method for producing large Carmichael numbers
- Dubner
- 1989
(Show Context)
Citation Context ...ime factors [30]. Wagstaff [37] used the so-called "universal forms" of Chernick [6] to find large Carmichael numbers. Woods and Huenemann [39] gave larger numbers using the extension Theore=-=m. Dubner [9]-=- found even larger Carmichael numbers using a modification of the universal forms, culminating in a 3710-digit number, and more recently in a 8060-digit number (email sent to the NMBRTHRY 19 rue Grang... |

1 |
una presumibile propriet`a caratteristica dei numeri primi
- Su
- 1950
(Show Context)
Citation Context ... N has r factors, then 2 r must divide N \Gamma 1. Moreover, if 5 divides p \Gamma 1 for many p's dividing N , then N must have many trailing zeroes, which we can spot at once. 5.3 Giuga's problem In =-=[11]-=-, Giuga asked whether there exist composite integers N such that N j 1 N \Gamma1 + 2 N \Gamma1 + \Delta \Delta \Delta + (N \Gamma 1) N \Gamma1 + 1: Equivalently, N must be a Carmichael number and sati... |

1 |
there exist composite numbers M for which kOE(M
- Do
- 1970
(Show Context)
Citation Context ...ur method. 5.2 Lehmer's problem In [22] Lehmer raised the following question: does there exist a composite integer N for which '(N) j N \Gamma 1? Such a number must be a Carmichael number. In view of =-=[26, 18, 7]-=- (see also [35]), such a number N must have a large number of prime factors (at least 14 and many more if 3 j N ). It is tempting to look at our numbers and test whether they satisfy the condition. Ho... |

1 |
On highly composite numbers
- Nicolas
- 1988
(Show Context)
Citation Context ...hat yield highly composite values ofs(recall that a highly composite number n is such that m ! n implies that m has fewer divisors than n; those numbers were first studied by Ramanujan [34]; see also =-=[29]-=-). 3.4 Remarks 3.4.1 Using outside primes While working out our ideas, we tried to modify the algorithm so as to produce Carmichael numbers all of whose prime factors where in S () but for one. More p... |

1 |
Larger Carmichael numbers
- Woods, Huenemann
- 1982
(Show Context)
Citation Context ... Pinch found the smallest Carmichael numbers with up to 20 prime factors [30]. Wagstaff [37] used the so-called "universal forms" of Chernick [6] to find large Carmichael numbers. Woods and =-=Huenemann [39]-=- gave larger numbers using the extension Theorem. Dubner [9] found even larger Carmichael numbers using a modification of the universal forms, culminating in a 3710-digit number, and more recently in ... |

1 |
Carmichael numbers with many prime factors
- Yorinaga
- 1980
(Show Context)
Citation Context ... = 8241; this is in agreement with our own calculations and that of Pinch) and by Pinch up to 10 15 (see [30]) and more recently up to 10 16 (email to the NMBRTHRY mailing list, April 1993). Yorinaga =-=[40, 41] has found-=- many Carmichael numbers using several methods including Chernick 's "extension theorem" (see Section 2). In particular, he found Carmichael numbers with 12 and up to 18 prime factors. Pinch... |

1 |
Searching for large Carmichael numbers. To appear in Sichuan Daxue Xuebao
- Zhang
- 1991
(Show Context)
Citation Context ...g Dickson pseudoprimes. mailing list, July 1994). Loh and Niebuhr [27, 28] have built numbers with up to 1; 101; 518 factors (this particular result was communicated to us by Keller). Recently, Zhang =-=[42]-=- has built a Carmichael number with 1305 prime factors and 8340 digits. The purpose of this paper is to analyze the method used by Loh and Niebuhr and to extend it to other classes of numbers. By one ... |