## 8087Re: [PrimeNumbers] Digest Number 638

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• Aug 2, 2002
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Hello,

> Did anybody look for primes of the form p#^2^n+1
> (or n!^2^n+1)? E.g., 1801#^16+1 is prime.

No, But!,
I did look at 2^n*n#-1 and have assembled my
fragments.
They have a good, neat looking form, but the form is
also flawed. For non-prime n, "proper" notation
would be 2^n*P(pi(n))#-1, ruining the repeating digits
effect in the presentation. When n is prime, the
notational collapse doesn't exist, so whatever these
primes should be called, n has to be prime for it to
be a perfect "whatever...".

There can be circumstances where taking the primorial
of n=composite can create several instances of the
same
number in an iterated run. With this form, the
multipler 2^n ensures each is distinct.
Thus, is it flawful, lawful, awful to say
2^16*16#-1 is prime?

Here's what I have found on these,
any one wants to take it farther,
be my guest.

-Dick

After n=16, they are all PRP's.
I never got around to testing them.

2^2*2#-1 is prime!
2^3*3#-1 is prime!
2^7*7#-1 is prime!
2^8*8#-1 is prime!
2^14*14#-1 is prime!
2^16*16#-1 is prime!
2^18*18#-1
2^20*20#-1
2^40*40#-1
2^42*42#-1
2^44*44#-1
2^53*53#-1
2^134*134#-1
2^154*154#-1
2^185*185#-1
2^187*187#-1
2^191*191#-1
2^197*197#-1
2^200*200#-1
2^201*201#-1
2^230*230#-1
2^235*235#-1
2^239*239#-1
2^244*244#-1
2^256*256#-1
2^282*282#-1
2^303*303#-1
2^358*358#-1
2^489*489#-1
2^536*536#-1
2^665*665#-1
2^684*684#-1
2^719*719#-1
2^1098*1098#-1
2^1204*1204#-1
2^1400*1400#-1
2^1516*1516#-1
2^1629*1629#-1
2^1903*1903#-1
2^1997*1997#-1
2^1999*1999#-1
2^2104*2104#-1
2^2477*2477#-1
2^3075*3075#-1
2^3676*3676#-1
2^3785*3785#-1
2^4115*4115#-1
2^5429*5429#-1
2^5808*5808#-1
2^6069*6069#-1
2^6276*6276#-1
2^9095*9095#-1
2^10423*10423#-1
2^10839*10839#-1
2^16181*16181#-1
2^17521*17521#-1
2^17734*17734#-1
2^20451*20451#-1
2^22560*22560#-1
2^29545*29545#-1
2^30069*30069#-1
2^33389*33389#-1
** Run Stopped at 34952

Here's the largest cofactor PRP's in the run
(2^12804*12804#-1)/15959
(2^15233*15233#-1)/263089
(2^18932*18932#-1)/116981
(2^27766*27766#-1)/1993067

And here are the caps of a couple Sophie Germains,
(no more of these at least to n=918).
2^(53-1)*53#-1 (digits:36 checksum:_8F526EDE_)
2^(201-1)*201#-1 (digits:143 checksum:_E1E2C9AF_)

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