ACAC Seminar Abstract

ACAC Seminar Abstract

ACAC Seminars

ACAC Seminar Abstract

Carmichael numbers of various shapes

Speaker: Florian Luca
Date, Time: Fri, 08 Mar 2013 13:30

Let k>=1 be an odd number. If N=2^n*k+1 is a Carmichael number, then n<2^{2*10^6*tau(k)^2*log(k)^2*omega(k)}. The proof of this result uses the Subspace Theorem. Further, the smallest odd k such that 2^n*k+1 is Carmichael for some n is k=27 (1729=2^6*27+1 is a Carmichael number). These results have obtained in joint work with J. Cilleruelo (Madrid) and A. Pizarro (Valparaiso). In the same spirit, in work in progress, we prove jointly with Banks, Finch, Pomerance and Stănică that the set {k odd : k=(N-1)/2^n for some Carmichael number N and some positive integer n} is of asymptotic density zero. These results together with some of the main steps of their proofs will be presented during the talk.

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