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Statement

The sets $\mathcal E$ (p. 704) and $\mathcal B$ (p. 705) are those defined on the Theorem 1 page: $E\in\mathcal E$ when $\pi(x,x^{1-E})\ge\gamma_1(E)\pi(x)$ for all $x\ge x_1(E)$, and $B\in\mathcal B$ when the lower bound (0.3) for primes in progressions holds for $d\le\min\{x^B,y/x^{1-B}\}$ outside the multiples of a bounded exceptional set.

Theorem 3 (printed p. 707): "For each $B\in\mathcal B$, $(0,B)\subset\mathcal E$."

The paper introduces it (p. 707) to show that, for Erdős's conjecture $C(x)\ge x^{1-\varepsilon}$, one need only assume $\mathcal B=(0,1)$, rather than both $\mathcal E=(0,1)$ and $\mathcal B=(0,1)$. It also remarks (p. 707) that the proofs of Theorems 1 and 3 need the definition of $\mathcal B$ only with $a=1$; and since its proof that $(0,5/12)\subset\mathcal B$ is effective, Theorem 3 gives computable $\gamma_1(E)$ and $x_1(E)$ for every $0<E<5/12$ (p. 707).

Source. W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703--722; Theorem 3 on p. 707, its proof in Section 5, pp. 720--721. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the surrounding remarks were read clause by clause on the page image of p. 707. The proof was not checked, and nothing here is independently reviewed.

Proof pointer

Section 5, pp. 720--721. Given $B\in\mathcal B$ and $0<\delta<B$, the proof removes one prime factor of each exceptional modulus, takes the remaining primes in $[x^{\delta/2},x^{\delta/2+\varepsilon}]$ with $\varepsilon=\delta^2/(20B)$, and counts pairs $(q,d)$ with $q\le x$ a prime $\equiv1\bmod d$ and $d\in[x^{B-\delta},x^B]$ a product of those primes; (0.3) bounds the count below, and each such $q$ has $q-1$ free of prime factors above $x^{1-B+\delta}$, which gives (0.1) for $E=B-\delta$.

Dependencies

The definition of $\mathcal B$ through (0.3) and Mertens' theorem.

Bears on

• Problem 1057: with Theorem 1, if $\mathcal B=(0,1)$ then $\mathcal E=(0,1)$ and $C(x)\ge x^{1-\varepsilon}$ for every $\varepsilon>0$ and all large $x$, which with $C(x)\le x$ is the problem's $C(x)=x^{1-o(1)}$ (p. 707). The hypothesis $\mathcal B=(0,1)$ is conjectural; the theorem does not decide the problem.