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Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
The pseudoprimes below 2 64
"... pseudoprime n. A backgammon prime (six consecutive occupied points) with one point missing. This term is an esoteric pun derived from number theory: a number that passes a certain kind of “primality test ” may be called a ‘pseudoprime ’ (all primes pass any such test, but so do some composite number ..."
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pseudoprime n. A backgammon prime (six consecutive occupied points) with one point missing. This term is an esoteric pun derived from number theory: a number that passes a certain kind of “primality test ” may be called a ‘pseudoprime ’ (all primes pass any such test, but so do some composite numbers), and any number that passes several is, in some sense, almost certainly prime. The hacker backgammon usage stems from the idea that a pseudoprime is almost as good as a prime: it will do the same job unless you are unlucky. The definition above includes primes as pseudoprimes. We do not: Definition. We write prp(n) to denote 2 n−1 ≡ 1 mod n (n is a probable prime). We write psp(n) (n is a pseudoprime) if n is also composite. � 1 A simple algorithm Algorithm 1 Enumerate all pseudoprimes below x (with repetition). 1 f o r (q ← 3; q ≤ x; q ← q + 2) { 2 f o r (q ′ ← 3; q ′ ≤ min(q, x/q) ; q ′ ← q ′ + 2) { 3 i f (2 qq ′ −1 ≡ 1 (mod qq ′)) { 4 enumerate qq ′}}} Algorithm 1 requires O(x ln 2 (x)) modular multiplications. 2 The goals • Develop better algorithms for enumerating pseudoprimes. • Extend Richard Pinch’s table: The pseudoprimes up to 10 13 [Pin00, Pin]—compiled circa 1994. • Check a conjecture on the density of pseudoprimes with two prime factors [Gal]. 3 The rank function Definition. Given odd q ∈ N, ρ(q) denotes the order of 2 in the multiplicative group modulo q. Equivalently, ρ(q) is the least r such that q  2 r − 1 (sometimes called the “rank of appearance ” of q). � Note: prp(n) ⇐ ⇒ ρ(n)  n − 1 ⇐ ⇒ n ≡ 1 mod ρ(n)