[Zhang 2006] *Notes on some new kinds of pseudoprimes*,
Mathematics of Computation, __75____:253 (2006)__, 451-460.

**A****BSTRACT**. 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* ∈{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},
…, *b** _{k}*, or a Syl-upsp(