[seqfan] wwww pseudoprimes

[Felix Fröhlich]

Dear SeqFans

this is related to what one could call wwww primes (Wall-Sun-Sun,
Wieferich, Wilson and Wolstenholme primes). I don't expect anything of this
to be added to the OEIS anytime soon, so these are just some thoughts.
Three of the sequences mentioned above are in the OEIS, they
are A001220, A007540 and A088164. The Wall-Sun-Sun primes are not in the
OEIS, since no such prime is currently known.

Each of those classes of primes can be defined via a congruence very
similar to a congruence satisfied by (almost) all primes p. The congruence
of the Wieferich primes is a stronger form of a special case of Fermat's
little theorem, the congruence of the Wilson primes is a stronger form of
the congruence in Wilson's theorem, the congruence of the Wolstenholme
primes is a stronger form of the congruence in Wolstenhome's theorem and
the congruence satisfied by the Wall-Sun-Sun primes is a stronger form of a
congruence satisfied by all odd prime numbers.

Now, what I actually want to get at is this: Given the above knowledge one
might wonder whether there are composites satisfying the stronger
congruences, i.e. whether there exist what could be called Wall-Sun-Sun
pseudoprimes, Wieferich pseudoprimes, Wilson pseudoprimes and Wolstenholme
pseudoprimes.

What I know is that Wieferich pseudoprimes do in fact exist (one can search
the OEIS for "wieferich pseudoprime" to find several sequences and comments
I made), although none appear to be known in the 'classical' Wieferich base
2. Also, it can be shown that any Wieferich pseudoprime to some base b must
be the product of base-b Wieferich primes.

Wilson's theorem implies that no Wilson pseudoprimes can exist. Regarding
the last two cases, not much appears to be known. A Wall-Sun-Sun
pseudoprime could be defined similar to a Wall-Sun-Sun prime, replacing the
Legendre symbol with the Kronecker symbol (see also the comment in
A113649). I believe that such a number would have to be a product of
Wall-Sun-Sun primes. A Wolstenholme pseudoprime seems quite unlikely to
exist. First, observe, that such a number would have to satisfy the
congruence from Wolstenholme's theorem. Only some prime powers and an
(apparently) quite scarce class of composites satisfy binomial(2*n-1, n-1)
== 1 (mod n) and I believe none are known that satisfy the congruence
modulo n^2.

Anyway, these are just some thoughts that maybe someone might enjoy to read.

Best regards
Felix Fröhlich