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Source. Theorem 2, p. 356, proved in §3, pp. 368–370, with Appendix A, pp. 371–372, 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

Theorem 2 (p. 356, quoted). "Infinitely many natural numbers are simultaneously Sierpiński, Riesel, and Carmichael. In fact, the number of them up to $x$ is $\gg x^{1/5}$ for all sufficiently large $x$."

A Sierpiński number is an odd natural $k$ with $2^nk+1$ composite for every $n\in\mathbb N$ (p. 355); a Riesel number is an odd natural $k$ with $2^nk-1$ composite for all $n\in\mathbb N$ (p. 356); a Carmichael number is a composite $N$ with $a^N\equiv a\pmod N$ for all integers $a$ (p. 355).

Proof pointer

§3, pp. 368–370; the proof of Theorem 2 itself is on pp. 369–370. As in Proposition 1, it suffices by the paper's Theorem 4 (p. 368, attributed to Matomäki) to find coprime $b,m$ with $b$ a quadratic residue modulo $m$ and every large member of $b\bmod m$ both Sierpiński and Riesel. The proof takes the collection (28) for the Sierpiński side and a second collection $\{(c_j,m_j;d_j,q_j)\}_{j=1}^M$ for the Riesel side, in which the $q_j$ are distinct primes, the classes $c_j\bmod m_j$ cover $\mathbb Z$, $q_j\mid2^{m_j}-1$, $q_j\mid2^{c_j}d_j-1$ and $d_j$ is a quadratic residue modulo $q_j$, with the two prime sets coprime; the Chinese remainder theorem then gives $b$ and $m$. The second collection is the one listed in Appendix A (pp. 371–372); the paper introduces Theorem 2 as proved with results of Matomäki and Wright coupled with an extensive computer search (p. 356).

Dependencies

Proposition 1's criterion and collection (28); Theorem 4 of the paper, attributed to K. Matomäki, Carmichael numbers in arithmetic progressions, J. Aust. Math. Soc. 94 (2013), no. 2, 268–275; the Appendix A tables. Read depth: claims checked; the statement and the reduction were read clause by clause, and the Appendix A tables were not checked.

Bears on

• Problem 1113: the Sierpiński property of every number the proof produces comes from the finite covering set of (28), so the theorem supplies no Sierpiński number without a finite covering set and neither proves nor disproves the problem.