## superpseudoprime

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• The sequence number with 9 prime numbre (a,b,c,m,n,k, x,y, z)   246241 ; 262657; 279073 ;  33975937 ; 209924353 ; 4261383649 ; 487824887233 ; 138991501037953
Message 1 of 22 , Sep 28, 2009
The sequence number with 9 prime numbre (a,b,c,m,n,k, x,y, z)

246241 ; 262657; 279073 ;  33975937 ; 209924353 ; 4261383649 ; 487824887233 ; 138991501037953 ; 24929060818265360451708193

a*b*c*m*n*k*x*y*z = n  is a pseudoprime  base 2

and  all composite factors of n are pesudoprime
we have 208 pseudoprime by 9 primes

ab     =  246241*262657 = 64676922337        =  pseudoprime
ac     =  68719214593                                       pseudoprime
am    =  8366268702817                                   pseudoprime
an     =  51691982607073                                 pseudoprime
ak     =  1049327371113409                              pseudoprime
ax    =   120122488057141153                          pseudoprime
ay    =   34225406207086584673                       pseudoprime
az    =   6138556864950480622989077152513    pseudoprime
bc    =   73300476961                                       pseudoprime
bm   =   8924017684609                                   pseudoprime
bn   =   55138100785921                                  pseudoprime
bk  =    1119282245095393                               pseudoprime
bx  =    128130621405958081                           pseudoprime
by  =    36507090688125621121                        pseudoprime
bz   =     6547792327343124780164318848801                               pseudoprime
cm  =     9481766666401                                                               pseudoprime
cn   =    58584218964769                                                              pseudoprime
ck   =    1189237119077377                                                           pseudoprime
cx   =    136138754754775009                                                       pseudoprime
cy   =     38788775169164657569                                                   pseudoprime
cz   =     6957027789735768937339560545089                                pseudoprime
mn  =     7132376592293761                                                          pseudoprime
mk  =     144784502391254113                                                       pseudoprime
mx  =     16574307635660512321                                                   pseudoprime
my  =      4722366482800925736961                                               pseudoprime
mz  =      846988199830552335989529107751841                            pseudoprime
nk  =      894568205401104097                                                      pseudoprime
nx  =      102406323829685485249                                                  pseudoprime
ny  =      29177700927891111969409                                               pseudoprime
nz  =      5233216963172006375136630160324129                           pseudoprime
kx  =      2078808998029975053217                                                pseudoprime
ky  =      592296109873099442630497                                             pseudoprime
kz  =      106232292155882567572000547769536257                        pseudoprime
xy  =      67803513320184824678154049                                         pseudoprime
xz  =      12161016282494898169024647192447199969                    pseudoprime
yz  =      3464927582597123310724584984291374048929                 pseudoprime
abcm   =      613251486300372864299137                                              pseudoprime
abcn    =      3789046980158167152145153                                            pseudoprime
abck    =      76916196790845133322670049                                           pseudoprime
abcx    =      8805035668330472737001476033                                       pseudoprime
abcy    =      2508738599163416594755412218753                                  pseudoprime
abcz    =     449959146053091093311287776772430639752993                p
abmn   =     461300166938020292886639457                                         pseudoprime
abmk  =       9364216016760333054532822081                                       pseudoprime
abmx =       1071975207741161047082948614177                                  pseudoprime
abmy  =       305428130254967320121131087397857                               pseudoprime
abmz =       54780590020796069994608704147266725025772417            pseudoprime
abnk  =       57857918345876673615721514689                                      pseudoprime
abnx  =      6623325853150240544585732106913                                   pseudoprime
abny   =       1887123896885426286038037012788833                             pseudoprime
abnz   =       338468367099746845497180716643175550080169473          pseudoprime
abkx   =       134450968119041482514963351008129                               pseudoprime
abky   =       38307889498769671576490641456711489                            pseudoprime
abkz =         6870777709447511120546434083811755068494672609         pseudoprime
abxy =   pseudoprime
abxz =   pseudoprime
abyz=    pseudoprime
acmn = pseudoprime
acmk = pseudoprime
acmx = pseudoprime
acmy=  pseudoprime
acmz=  pseudoprime
acnk =  pseudoprime
acnx=   pseudoprime
acny =  pseudoprime
acnz =  pseudoprime
ackx  = pseudoprime
acky  = pseudoprime
ackz  = pseudoprime
acxy  = pseudoprime
acxz =  pseudoprime
acyz  = pseudoprime
amnk =    pseudoprime
amnx =    pseudoprime
amny  =  pseudoprime
amnz =   pseudoprime
amkx  =  pseudoprime
amky  =  pseudoprime
amkz  =  pseudoprime
amxy  =  pseudoprime
amxz  =  pseudoprime
amyz =   pseudoprime
ankx  =  pseudoprime
anky  =  pseudoprime
ankz =  pseudoprime
anxy  =  pseudoprime
anxz  =   pseudoprime
anyz   =  pseudoprime
akxy  =   pseudoprime
akyz  =   pseudoprime
axyz =    pseudoprime
abmnk  =   pseudoprime
abmkx =    pseudoprime
abmxy =    pseudoprime
abmyz  =   pseudoprime
abnkx  =    pseudoprime
abnxy  =    pseudoprime
abnyz  =    pseudoprime
abkxy  =    pseudoprime
abkyz  =    pseudoprime
abxyz  =    pseudoprime
abcmn  =   pseudoprime
abcmk =      pseudoprime
abcmx =     pseudoprime
abcmy  =    pseudoprime
abcmz   =   pseudoprime
acmnk  =    pseudoprime
acmnx =    pseudoprime
acmny  =   pseudoprime
acmnz  =    pseudoprime
amnkx =    pseudoprime
amnky  =   pseudoprime
amnkz  =    pseudoprime
ankxy  =     pseudoprime
ankxz  =     pseudoprime
ankyz =      pseudoprime
akxyz  =     pseudoprime
abcmnk =     pseudoprime
abcmnx =    pseudoprime
abcmny  =   pseudoprime
abcmnz  =   pseudoprime
abcmkx  =   pseudoprime
abcmky  =   pseudoprime
abcmkz =    pseudoprime
abcmxy  =   pseudoprime
abcmyz  =   pseudoprime
abcnxy  =    pseudoprime
abcnyz =     pseudoprime
abckxy  =    pseudoprime
abckyz  =    pseudoprime
abcxyz =     pseudoprime
abmnkx  =  pseudoprime
abmnky  =  pseudoprime
abmnkz  =  pseudoprime
abnkxy  =   pseudoprime
abnkxz  =   pseudoprime
abkxyz =      pseudoprime
abcmnkx =   pseudoprime
abcmnky =   pseudoprime
abcmnkz =   pseudoprime
abcmnxy  =  pseudoprime
abcmny   =   pseudoprime
cmnkxyz =   pseudoprime
bmnkxyz =   pseudoprime
bcnkxyz =    pseudoprime
bcmkxyz =   pseudoprime
bcmnxyz =   pseudoprime
bcmnkyz =   pseudoprime
bcmnkxz =   pseudoprime
amnkxyz =   pseudoprime
acnkxyz  =   pseudoprime
acmkxyz =     pseudoprime
acmnxyz  =   pseudoprime
acmnkyz =    pseudoprime
acmnkxz =    pseudoprime
acmnkxy =    pseudoprime
abnkxyz  =  pseudoprime
abmkxyz =  pseudoprime
abmnxyz =  pseudoprime
abmnkyz =  pseudoprime
abmnkxz =  pseudoprime
abmnkxy =  pseudoprime
abckxyz =  pseudoprime
abcnxyz =  pseudoprime
abcnkyz =  pseudoprime
abcnkxz =  pseudoprime
abcnkxy =  pseudoprime
abcmxyz =  pseudoprime
abcmkyz =  pseudoprime
abcmkxz =  pseudoprime
abcmkxy =  pseudoprime
abcmnyz =  pseudoprime
abcmnxz =  pseudoprime
abcmnxy =  pseudoprime
abcmnkz =  pseudoprime
abcmnky =  pseudoprime
abcmnkx =  pseudoprime
abcmnkxy =  pseudoprime
abcmnkxz =  pseudoprime
abcmnkyz =  pseudoprime
abcmnxyz =  pseudoprime
abcmkxyz =  pseudoprime
abcnkxyz  =  pseudoprime
abmnkxyz =  pseudoprime
acmnkxyz =  pseudoprime
bcmnkxyz =  pseudoprime

a*b*c*m*n*k*x*y*z = 927278484441774426341627658843620260247978492902026521810901981921188130379449817967064027073 is a pesudoprime base 2

conclusion :  n is a psudoprime with 9 prime factor
all  composite  factor are pseudoprime.

pour plus d’informations consultez la page
thanks

[Non-text portions of this message have been removed]
• sory for all if the message is not in order you can Visithttp://groups.google.co.ma/group/-sequence-of-prime-numbers-/web/superpseudpprime?hl=fr   thanks
Message 2 of 22 , Sep 29, 2009
sory for all if the message is not in order

thanks

[Non-text portions of this message have been removed]
• ... 2^9 - 10 = 502 products are base-2 pseudoprimes: {a=[246241,262657,279073,33975937,209924353,4261383649,
Message 3 of 22 , Sep 29, 2009
Lio David <maths_forall@...> wrote:

> sory for all if the message is not in order

\\ 2^9 - 10 = 502 products are base-2 pseudoprimes:

{a=[246241,262657,279073,33975937,209924353,4261383649,
487824887233,138991501037953,24929060818265360451708193];
c=0;for(k=1,511,b=binary(512+k);p=prod(k=1,9,a[k]^b[k+1]);
if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)}

502

David
• ... nice find - congrats to the original poster ! ... Your duplicate usage of the loop variable k is original but maybe a little confusing... note that
Message 4 of 22 , Sep 29, 2009
On Tue, Sep 29, 2009 at 6:20 AM, djbroadhurst wrote:
> \\ 2^9 - 10 = 502 products are base-2 pseudoprimes:

nice find - congrats to the original poster !

> {a=[246241,262657,279073,33975937,209924353,4261383649,
> 487824887233,138991501037953,24929060818265360451708193];
> c=0;for(k=1,511,b=binary(512+k);p=prod(k=1,9,a[k]^b[k+1]);
> if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)}

Your duplicate usage of the loop variable k is original but maybe a little confusing... note that bittest() can replace the "binary"-hack:

c=0;for(k=1,511,p=prod(j=1,9,a[j]^bittest(k,j-1));if(!isprime(p)&&Mod(2,p)^p==2,c+=1));print(c)

or simply:

sum(k=1,2^#a-1, !isprime(p=prod(j=1,#a,a[j]^bittest(k,j-1))) && Mod(2,p)^p==2)

PS: are all of these primes in
http://www.research.att.com/~njas/sequences/A104885
Primes whose logarithms are known to possess binary BBP formulas.

and if so, is there a simple explanation ?

Maximilian
• ... Interesting question! No idea, my end. The most relevant OEIS sequence is http://www.research.att.com/~njas/sequences/A050217 leading to
Message 5 of 22 , Sep 29, 2009
--- In primenumbers@yahoogroups.com, "maximilian_hasler" <maximilian.hasler@...> wrote:

> PS: are all of these primes in
> http://www.research.att.com/~njas/sequences/A104885
> Primes whose logarithms are known to possess binary BBP formulas.

Interesting question! No idea, my end.

The most relevant OEIS sequence is
http://www.research.att.com/~njas/sequences/A050217
http://en.wikipedia.org/wiki/Super-Poulet_number
which makes it clear that Cunningham project
factors are useful here.

http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt
> 108 33975937.138991501037953
> 216 209924353.4261383649.24929060818265360451708193

Here are the indices of that set of 9:

{a=[246241,262657,279073,33975937,209924353,4261383649,
487824887233,138991501037953,24929060818265360451708193];
for(k=1,9,p=a[k];for(j=1,1000,
if(Mod(2,p)^j==1,print([k,j]);break)))}

[1, 108]
[2, 27]
[3, 108]
[4, 216]
[5, 432]
[6, 432]
[7, 144]
[8, 216]
[9, 432]

So the "Super Poulet" number is simply
a divisor of 2^432-1, in this case.

David
• ... thanks for pointing out these. ... I also remarked this - by a simple google search for the larger numbers.... Just for information, I consider submitting
Message 6 of 22 , Sep 29, 2009

thanks for pointing out these.

> which makes it clear that Cunningham project
> factors are useful here.

I also remarked this - by a simple google search for the larger numbers....

Just for information, I consider submitting

294409, 1398101, 1549411, 1840357,12599233, 13421773, 15162941, 15732721, 28717483, 29593159, 61377109, 66384121, 67763803, 74658629, 78526729, 90341197, 96916279, 109322501, 135945853, 153369061, 157010389, 163442551, 206453509, 221415781, 231927781, 271682651, 351593899, 367632301, 434042801, 457457617, 464955857, 491738801, 516045197, 536870911, 604611019, 611097401, 612006253, 630622753, 762278161

to the OEIS, as
A165777 Super-pseudoprimes (to base 2)
COMMENT:
A super-pseudoprime (to base b) is a pseudoprime to base b whose composite divisors are again pseudoprimes to base b, and which is not a semiprime (so that it has at least one composite proper divisor).

All super-pseudoprimes below 10^9 are listed above, they all have omega(n)=3. (Actually I must verify that there are none with omega=2 and bigomega>2 - by error my script only considered those with omega>2 instead of bigomega>2.)

What is the least super-pseudoprime with bigomega > 3 ?

Maximilian
• ... The OP pointed to a claim that le plus grand nombre superpseudoprime connu est 92727848.....64027073 avec 9 factors premier. The plus grand ... connu
Message 7 of 22 , Sep 29, 2009
"maximilian_hasler" <maximilian.hasler@...> wrote:

> nice find - congrats to the original poster !

The OP pointed to a claim that
"le plus grand nombre superpseudoprime connu est
92727848.....64027073 avec 9 factors premier."

The "plus grand ... connu" claim is rather weird:
to make a bigger one, we simply multiply the
original number by 433.

The construction is very simple, given the
Cunningham project. We multiply the listed prime
factors of 2^432 - 1 that are congruent
to 1 mod 432. Then we have a so-called "Super Poulet"
number, N, with 10 (not 9) distinct prime divisors.

Proof: By construction, every divisor d|N satisfies
d = 1 mod 432 and 2^432 = 1 mod d.
Hence 2^d = 2 mod d and we are done.

More like Cunningham fried chicken than Super Poulet?

David
• ... Here s some more fast food from the good Colonel: 70171342151*Phi(410,2)*Phi(1025,4) is a 541-digit Super Poulet number with 2^16 - 17 = 65519 pseudoprime
Message 8 of 22 , Sep 29, 2009

> More like Cunningham fried chicken than Super Poulet?

Here's some more fast food from the good Colonel:

70171342151*Phi(410,2)*Phi(1025,4)
is a 541-digit Super Poulet number with
2^16 - 17 = 65519 pseudoprime divisors

Incidentally, Cunningham really was a colonel. See
http://www.nature.com/nature/journal/v89/n2213/abs/089086c0.html

David
• ... The Wieferich constructions 1093^2 * 4733 3511^2 * 1969111 are Super Poulet numbers that are not square-free. David
Message 9 of 22 , Sep 29, 2009
"maximilian_hasler" <maximilian.hasler@...> wrote:

> I must verify that there are none with
> omega=2 and bigomega>2

The Wieferich constructions
1093^2 * 4733
3511^2 * 1969111
are Super Poulet numbers that are not square-free.

David
• ... (4^21-1)/21^2 = 9972894583 == 43*127*337*5419 with certainty. The least with bigomega 4 may be Phi(51,4)/11119 = 1264022137981459 =
Message 10 of 22 , Sep 29, 2009

> What is the least super-pseudoprime with bigomega > 3 ?

(4^21-1)/21^2 = 9972894583 == 43*127*337*5419
with certainty.

The least with bigomega > 4 may be
Phi(51,4)/11119 = 1264022137981459 = 103*307*2143*2857*6529
but I am not sure of that.

David
• ... The largest known Super Poulet is quite large: (4^(2^43112609 - 1) - 1)/3 David
Message 11 of 22 , Sep 30, 2009
Lio David <maths_forall@...> wrote:

> There's Not Sense for the search

The largest known Super Poulet is quite large:

(4^(2^43112609 - 1) - 1)/3

David
• yes the superpoulet is the cousin of  the superpseudoprime lol a simple way to make the superpoulet n is even numbre   factorize 2^n-1  and select the
Message 12 of 22 , Sep 30, 2009
yes the superpoulet is the cousin of  the superpseudoprime lol

a simple way to make the superpoulet
n is even numbre

factorize 2^n-1  and select the factor witch k = 1 mod n

example n = 100
2^100-1 = 5*5*5*11*31*41*101*251*601*1801*4051*8101*268501

so the test 101*601      is superpoulet
101*1801    is superpoulet
601*1801    is superpoulet

rachid

[Non-text portions of this message have been removed]
• ... Bien sûr. Mais vous êtes venu ici trop tard: nous avons déjà décidé tout cela, avec des exemples énormes. Amitiés David
Message 13 of 22 , Sep 30, 2009
Lio David <maths_forall@...> wrote:

> a simple way to make the superpoulet
> n is even numbre
> factorize 2^n-1 and select the factors with
> k = 1 mod n

Bien sûr. Mais vous êtes venu ici trop tard:
nous avons déjà décidé tout cela, avec des
exemples énormes.

Amitiés

David
• ... Bonsoir à tous, bonsoir David, Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l on discute majoritairement
Message 14 of 22 , Sep 30, 2009
>
> Lio David <maths_forall@> wrote:
>
> > a simple way to make the superpoulet
> > n is even numbre
> > factorize 2^n-1 and select the factors with
> > k = 1 mod n
>
> Bien sûr. Mais vous êtes venu ici trop tard:
> nous avons déjà décidé tout cela, avec des
> exemples énormes.
>
> Amitiés
>
> David
>

Bonsoir à tous, bonsoir David,

Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.

On y parle de poulets (et autres volailles ?) avec des super pouvoirs. Espérons juste que ces drôles de gallinacés ne soient pas porteurs du virus H5N1 !

;-)

Jean-Louis
• ... bah moi ce qui m importe surtout c est le SNR du groupe (auf english: What matters for me is the Signal-to-Noise ratio of the group more thant the
Message 15 of 22 , Sep 30, 2009
j_chrtn wrote:
>> Bien sûr. Mais vous êtes venu ici trop tard:
>> nous avons déjà décidé tout cela, avec des
>> exemples énormes.
>>
>> Amitiés
>>
>> David
>
> Bonsoir à tous, bonsoir David,
>
> Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.

bah moi ce qui m'importe surtout c'est le SNR du groupe

(auf english: What matters for me is the Signal-to-Noise ratio of the group
more thant the language)

> Jean-Louis
yg
• ... s/under/oder/ ? grüsse, ... yg
Message 16 of 22 , Sep 30, 2009
Norman Luhn wrote:
> Na was denn nun? Englisch, Franzoesich under Deutsch ?
s/under/oder/ ?

grüsse,

> Hehe
yg
• Na was denn nun? Englisch, Franzoesich under Deutsch ? Hehe ... Von: Yann Guidon Betreff: Re: [PrimeNumbers] Re: superpseudoprime An:
Message 17 of 22 , Sep 30, 2009
Na was denn nun? Englisch, Franzoesich under Deutsch ?

Hehe

--- Yann Guidon <whygee@...> schrieb am Mi, 30.9.2009:

Von: Yann Guidon <whygee@...>
Datum: Mittwoch, 30. September 2009, 22:26

j_chrtn wrote:

>> Bien sûr. Mais vous êtes venu ici trop tard:

>> nous avons déjà décidé tout cela, avec des

>> exemples énormes.

>>

>> Amitiés

>>

>> David

>

> Bonsoir à tous, bonsoir David,

>

> Cela fait plaisir de pouvoir lire quelques mots dans la langue de Molière sur ce forum où l'on discute majoritairement dans celle de Shakespeare.

bah moi ce qui m'importe surtout c'est le SNR du groupe

(auf english: What matters for me is the Signal-to-Noise ratio of the group

more thant the language)

> Jean-Louis

yg

[Non-text portions of this message have been removed]
• ... Pour être tout à fait complet, il me faut ajouter que le plus grand Poulet connu : N = (4^(2^43112609 - 1) - 1)/3 est vraiment un super-Poulet, parce
Message 18 of 22 , Sep 30, 2009
"j_chrtn" <j_chrtn@...> wrote:

> Cela fait plaisir de pouvoir lire quelques mots dans
> la langue de Molière sur ce forum où l'on discute
> majoritairement dans celle de Shakespeare.

> On y parle de poulets (et autres volailles ?) avec des
> super pouvoirs.

Pour être tout à fait complet, il me faut ajouter que
"le plus grand Poulet connu":
N = (4^(2^43112609 - 1) - 1)/3
est vraiment un super-Poulet, parce que chaque diviseur
d|N possède la propriété que 2^d = 2 mod d,
et il y'en a, au moins, 4 tels diviseurs.
Mais on ne sait point si N est un super-pseudopremier,
parce que il nous reste les deux
possibilités (tout à fait éloignées) que
2^(2^43112609 - 1) - 1
est un permier de Mersenne et que
(2^(2^43112609 - 1) + 1)/3
est un premier de Wagstaff.

Comment pouvons nous exclure au moins
une de ces possibilités bizarres?

Il me semble extrêmement difficile, à ce moment,
parce que chaque premier p|N est nécessairement
plus grand que le plus grand premier connu.

"Plus l'obstacle était grand, plus fort fut le désir."

David (en l'esprit d'une entente cordiale)
• ... Ich meine etwas ganz einfach: kannst Du zeigen dass 2^(2^43112609 - 1) - 1 ist keine Primzahl, oder dass (2^(2^43112609 - 1) + 1)/3 ist keine Primzhal?
Message 19 of 22 , Sep 30, 2009
Norman Luhn <nluhn@...> wrote:

> Na was denn nun? Englisch, Franzoesich under Deutsch ?

Ich meine etwas ganz einfach: kannst Du zeigen dass
2^(2^43112609 - 1) - 1
ist keine Primzahl, oder dass
(2^(2^43112609 - 1) + 1)/3
ist keine Primzhal?

Schwierig, nicht wahr?

Alles Gute!

David
• ... Oh dear, I am out of my linguistic depth: I forgot that Norman speaks backwards :-) ... Translation: Can you show that 2^(2^43112609 - 1) - 1 is not prime,
Message 20 of 22 , Sep 30, 2009

> kannst Du zeigen dass
> 2^(2^43112609 - 1) - 1
> ist keine Primzahl, oder dass
> (2^(2^43112609 - 1) + 1)/3
> ist keine Primzahl?

Oh dear, I am out of my linguistic depth:
I forgot that Norman speaks backwards :-)

Correction:

> kannst Du zeigen dass
> 2^(2^43112609 - 1) - 1
> keine Primzahl ist, oder dass
> (2^(2^43112609 - 1) + 1)/3
> keine Primzahl ist?

Translation:

Can you show that
2^(2^43112609 - 1) - 1
is not prime, or that
(2^(2^43112609 - 1) + 1)/3
is not prime?

If (like me) you cannot, then you also
cannot show that the proven Super Poulet
4^(2^43112609 - 1) - 1)/3
is also a super-pseudoprime,
according to the strict definition of Maximilian.

• 86225219*5259738299*5949540043*12482997260297*(2^43112609-1) is the largest known completely factorized superpseudoprime, discovered by Edson Smith
Message 21 of 22 , Oct 1, 2009
86225219*5259738299*5949540043*12482997260297*(2^43112609-1)
is the largest known completely factorized superpseudoprime,
discovered by Edson Smith
http://primes.utm.edu/bios/page.php?id=1498
and Alex Kruppa
http://www.mersenneforum.org/showpost.php?p=142690&postcount=712

Puzzle: Find another superpseudoprime with at least
a million decimal digits and precisely 32 divisors.

Hint: This may be done by judicious googling.

• A base-b superpseudoprime is a non-semiprime composite number all of whose composite divisors are base-b pseudoprimes.
Message 22 of 22 , Oct 2, 2009
A base-b superpseudoprime is a non-semiprime composite
number all of whose composite divisors are base-b pseudoprimes.

1340753*2011129*803278043*(89^11971-1)/88 is a gigantic
base-89 superpseudoprime with precisely 11 composite divisors.

Puzzle 89: For a base with 89 > b > 2, find a gigantic
base-b superpseudoprime with precisely 26 composite divisors.

Hint: For the meat, see http://aruljohn.com/Bible/kjv/luke/12/42