Two contradictory conjectures concerning Carmichael numbers

Two contradictory conjectures concerning Carmichael numbers

Erdős conjectured that there are x1−o(1) Carmichael numbers up to x, whereas Shanks was skeptical as to whether one might even find an x up to which there are more than √ x Carmichael numbers. Alford, Granville and Pomerance showed that there are more than x2/7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdős must be correct...

On Two Conjectures concerning Convex Curves

In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP . Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them....

Carmichael numbers and pseudoprimes

We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...

Higher-order Carmichael numbers

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...

Pseudoprimes and Carmichael Numbers