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Source. Theorem 1, p. 356, with the reformulation of §2.1, p. 357, of William Banks, Carrie Finch, Florian Luca, Carl Pomerance and Pantelimon Stănică, Sierpiński and Carmichael numbers, Transactions of the American Mathematical Society 367 (2015), no. 1, 355–376, as identified on the source card.

Statement

A Carmichael number is a composite $N$ with $a^N\equiv a\pmod N$ for every integer $a$ (p. 355).

Theorem 1 (p. 356, quoted). "Almost all odd natural numbers $k$ have the property that $2^nk+1$ is not a Carmichael number for any $n\in\mathbb N$."

The paper makes "almost all" precise in §2.1 (p. 357): with

$$\mathcal C(x)=\{\text{odd }k\in(x/2,x]:\ 2^nk+1\text{ is Carmichael for some }n\},$$

Theorem 1 is the statement that $\lvert\mathcal C(x)\rvert=o(x)$ as $x\to\infty$. Summing over dyadic ranges, the exceptional odd $k\le x$ number $o(x)$, so the exceptional set has asymptotic density zero (an observation of this page; the paper reads "almost all" this way). The paper notes (p. 356) that consequently the set of odd parts $2^{-v_2(n-1)}(n-1)$ of Carmichael numbers $n$ has asymptotic density zero. The theorem gives no rate.

Proof pointer

§2, pp. 357–368. The proof removes from $\mathcal C(x)$ a chain of negligible sets $\mathcal C_1(x),\ldots,\mathcal C_{13}(x)$, each of size $o(x)$, sorted by the least exponent $n_0(k)$ giving a Carmichael value. Lemma 1 (p. 358) counts the $k$ with a Carmichael value $2^nk+1$, $n\le X$, divisible by a member of a family $\mathcal Q$ by $xX\sum_{q\in\mathcal Q}q^{-1}+X\lvert\mathcal Q\rvert$. Small $n_0(k)$ (§2.2, p. 358) are handled by Pomerance's upper bound for the counting function of Carmichael numbers. Medium $n_0(k)$ (§2.3, pp. 359–362) use Korselt's criterion, which forces every prime factor of a Carmichael $2^nk+1$ to be $2^md+1$ with $d\mid k$, together with Lemma 1. Large $n_0(k)$ (§§2.4–2.5, pp. 362–368) use a pigeonhole bound (12) for prime factors, Lemma 2 (p. 363, essentially [11, Lemma 7] of Cilleruelo, Luca and Pizarro-Madariaga, resting on a quantitative Subspace Theorem, $S$-unit bounds and linear forms in logarithms), the exponent bound (2) (p. 356) from the same paper [11], which restricts $n$ to $n\le\exp((\log x)^4)$, and a Brun-sieve count of the type III primes (p. 368).

Dependencies

Lemmas 1 and 2 of the paper; the bound (2) and Lemmas 2, 3 and 7 of J. Cilleruelo, F. Luca and A. Pizarro-Madariaga, Carmichael numbers in the sequence $\{2^nk+1\}_{n\ge1}$, Math. Comp. (to appear when the paper was printed); C. Pomerance, On the distribution of pseudoprimes, Math. Comp. 37 (1981), 587–593; G. Tenenbaum, Compositio Math. 51 (1984), 243–263, for integers with a divisor in a given interval. Read depth: claims checked; the statement and §2.1 were read clause by clause, the proof for its structure only.

Bears on

• Problem 1113: only through Corollary 1, which combines the theorem with the positive lower density of Sierpiński numbers. The theorem itself concerns Carmichael values for almost every odd $k$ and says nothing about covering sets; it neither proves nor disproves the problem.