Notes on some new kinds of pseudoprimes

J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031-1037) some new kinds of pseudoprimes, called Sylow $p$-pseudoprimes and elementary Abelian $p$-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow $p$-pseudoprime to two bases only, where $p=2$ or $3$.

In this paper, in contrast to Browkin's examples, we give facts and examples which are unfavorable for Browkin's observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow $2$-pseudoprimes to the same bases. In Section 3, we give examples of Sylow $p$-pseudoprimes to the first several prime bases for the first several primes $p$. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow $p$-pseudoprime to the same bases for all $p\in\{2,3,5,7,11,13\}$. In Section 4, we define $n$ to be a $k$-fold Carmichael Sylow pseudoprime, if it is a Sylow $p$-pseudoprime to all bases prime to $n$ for all the first $k$ smallest odd prime factors $p$ of $n-1$. We find and tabulate all three $3$-fold Carmichael Sylow pseudoprimes $<10^{16}$. In Section 5, we define a positive odd composite $n$ to be a Sylow uniform pseudoprime to bases $b_1,\ldots,b_k$, or a Syl-upsp $(b_1,\ldots,b_k)$ for short, if it is a Syl$_p$-psp $(b_1,\ldots,b_k)$for all the first $\omega(n-1)-1$ small prime factors $p$ of $n-1$, where $\omega(n-1)$ is the number of distinct prime factors of $n-1$. We find and tabulate all the 17 Syl-upsp$(2,3,5)$'s $<10^{16}$ and some Syl-upsp $(2,3,5,7,11)$'s $<10^{24}$. Comparisons of effectiveness of Browkin's observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6.