Persistent Link:
http://hdl.handle.net/10150/289053
Title:
On the infinitude of elliptic Carmichael numbers
Author:
Ekstrom, Aaron Todd
Issue Date:
1999
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
For any elliptic curve E with complex multiplication by an order in K=Q(√-d), a point Q of infinite order on E, and any prime p with gcd{Δ(E),p} = 1, (-dǀp) = -1 , we have that [p+1]·Q = O (mod p) where Δ(E) is the discriminant of E, O is the point at infinity and calculations are done using the addition law for E. Any composite number p which satisfies these conditions for any rational point on any CM elliptic curve over Q with discriminant prime to p is called an elliptic Carmichael number. For our main result, we modify the techniques of Alford, Granville and Pomerance to show there exist infinitely many elliptic Carmichael numbers under the assumption that the smallest prime congruent to -1 modulo q is at most q exp[(log q)¹⁻ᵋ]. (Note that this assumption is much weaker than current conjectures about the smallest prime in an arithmetic progression.) We modify the construction of Chernick to provide many examples of elliptic Carmichael numbers. We also show that elliptic Carmichael numbers are squarefree, and modify the techniques of Pomerance, Selfridge, and Wagstaff to prove that the number of elliptic Carmichael numbers up to x with exactly k factors is at most x⁽²ᵏ⁻¹⁾/⁽²ᵏ⁾⁺ᵋ for large enough x. Finally, we prove there are no strong elliptic Carmichael numbers, an analogue of Lehmer's result about strong Carmichael numbers.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Thakur, Dinesh S.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleOn the infinitude of elliptic Carmichael numbersen_US
dc.creatorEkstrom, Aaron Todden_US
dc.contributor.authorEkstrom, Aaron Todden_US
dc.date.issued1999en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractFor any elliptic curve E with complex multiplication by an order in K=Q(√-d), a point Q of infinite order on E, and any prime p with gcd{Δ(E),p} = 1, (-dǀp) = -1 , we have that [p+1]·Q = O (mod p) where Δ(E) is the discriminant of E, O is the point at infinity and calculations are done using the addition law for E. Any composite number p which satisfies these conditions for any rational point on any CM elliptic curve over Q with discriminant prime to p is called an elliptic Carmichael number. For our main result, we modify the techniques of Alford, Granville and Pomerance to show there exist infinitely many elliptic Carmichael numbers under the assumption that the smallest prime congruent to -1 modulo q is at most q exp[(log q)¹⁻ᵋ]. (Note that this assumption is much weaker than current conjectures about the smallest prime in an arithmetic progression.) We modify the construction of Chernick to provide many examples of elliptic Carmichael numbers. We also show that elliptic Carmichael numbers are squarefree, and modify the techniques of Pomerance, Selfridge, and Wagstaff to prove that the number of elliptic Carmichael numbers up to x with exactly k factors is at most x⁽²ᵏ⁻¹⁾/⁽²ᵏ⁾⁺ᵋ for large enough x. Finally, we prove there are no strong elliptic Carmichael numbers, an analogue of Lehmer's result about strong Carmichael numbers.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorThakur, Dinesh S.en_US
dc.identifier.proquest9957966en_US
dc.identifier.bibrecord.b40143417en_US
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