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.
