A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper, we present two new results.

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We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log \log n)$ time and $O(\sqrt{n}\log n)$ bits of space.

The second result solves an open problem given by Paul Pritchard in 1994.

We give an upper bound for the number of elliptic Carmichael numbers $n\,\le \,x$ that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non $\text{CM}$). We also discuss several possible further improvements.

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

Let ${{T}_{n}}$ denote the $n$-th Chebyshev polynomial of the first kind, and let ${{U}_{n}}$ denote the $n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res$({{T}_{m}},\,{{T}_{n}})$. Similarly, we give a formula for res$({{U}_{m}},\,{{U}_{n}})$.

Assuming a conjecture intermediate in strength between one of Chowla and one of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m≥1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

Browkin [‘Some new kinds of pseudoprimes’, Math. Comp. 73 (2004), 1031–1037] gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprimes to two bases only, where p=2 or 3. In contrast to Browkin’s examples, Zhang [‘Notes on some new kinds of pseudoprimes’, Math. Comp. 75 (2006), 451–460] gave facts and examples which are unfavorable for Browkin’s observation on detecting compositeness of odd composite numbers. In particular, Zhang gave a Sylowp-pseudoprime (with 27 decimal digits) to the first 6 prime bases for all the first 6 primes p, and conjectured that for any k≥1, there would exist Sylow p-pseudoprimes to the first k prime bases for all the first k primes p. In this paper we tabulate 27 Sylow p-pseudoprimes less than 1036 to the first 7 prime bases for all the first 7 primes p (two of which are Sylow p-pseudoprimes to the first 7 prime bases for all the first 8 primes p). We describe the procedure for finding these numbers. The main tools used in our method are the cubic residue characters and the Chinese remainder theorem.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D\,>\,3705$, there is always at least one prime $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$.