## Re: [PrimeNumbers] Dead Pigeons (was perimetric prime sieve/generator = 2(p^(n-k)+p^k) )

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• Bill Hmmm ... with reference to your conjecture below: Let us set e=2(p^(n-k)+p^k) then: For p=11, n=4, k=2 we have e=484 giving e-1=7*23 and e+1=5*97 For
Message 1 of 9 , Jun 30, 2001
Bill

Hmmm ... with reference to your conjecture below:

Let us set e=2(p^(n-k)+p^k) then:

For p=11, n=4, k=2 we have e=484 giving e-1=7*23 and e+1=5*97

For p=13, n=3, k=1 we have e=364 giving e-1=3*11^2 and e+1=5*73

For p=13, n=3, k=2 we have e=364 giving e-1=3*11^2 and e+1=5*73

For p=17, n=4, k=2 we have e=1156 giving e-1=3*5*7*11 and e+1=13*89

For p=17, n=5, k=2 we have e=10404 giving e-1=101*103 and e+1=5*2081

and so on .... in general an infinite pile of dead pigeons!

These kinds of conjectures are best first validated with a few simple
lines of Mathematica or Maple, for example:

Do[p=Prime[pi];
e=2(p^(n-k)+p^k);
If[!(PrimeQ[e-1]||PrimeQ[e+1]),
Print["p=",p," n=",n," k=",k," e=",e," e+1=",FactorInteger[e-1]," e-1=",FactorInteger[e+1]]];
,{pi,2,8},{n,1,pi-1},{k,1,n-1}];

Regards

Alan Powell
Pigeon slayer

At 03:07 AM 6/30/01, you wrote:
>take a prime number (p), raise it to any power (n),
>make a rectangle out of that number (with sides
>p^(n-k) and p^k). Calculate the perimeter of that
>rectangle. Add or subtract 1. At least one of these
>will be a prime number ... I think. At any rate, all
>the primes may be generated this way ... I think.
• Dear Alan et al, can you find any for me that don t work for e=2(2^(n-k)+2^k)? Bill ... ===== Bill Krys Email: billkrys@yahoo.com Toronto, Canada (currently:
Message 2 of 9 , Jun 30, 2001
Dear Alan et al,

can you find any for me that don't work for
e=2(2^(n-k)+2^k)?

Bill

--- Alan Powell <powella@...> wrote:
> Bill
>
> Hmmm ... with reference to your conjecture below:
>
> Let us set e=2(p^(n-k)+p^k) then:
>
> For p=11, n=4, k=2 we have e=484 giving e-1=7*23
> and e+1=5*97
>
> For p=13, n=3, k=1 we have e=364 giving e-1=3*11^2
> and e+1=5*73
>
> For p=13, n=3, k=2 we have e=364 giving e-1=3*11^2
> and e+1=5*73
>
> For p=17, n=4, k=2 we have e=1156 giving
> e-1=3*5*7*11 and e+1=13*89
>
> For p=17, n=5, k=2 we have e=10404 giving
> e-1=101*103 and e+1=5*2081
>
> and so on .... in general an infinite pile of dead
> pigeons!
>
>
> These kinds of conjectures are best first validated
> with a few simple
> lines of Mathematica or Maple, for example:
>
> Do[p=Prime[pi];
> e=2(p^(n-k)+p^k);
> If[!(PrimeQ[e-1]||PrimeQ[e+1]),
> Print["p=",p," n=",n," k=",k," e=",e,"
> e+1=",FactorInteger[e-1],"
> e-1=",FactorInteger[e+1]]];
> ,{pi,2,8},{n,1,pi-1},{k,1,n-1}];
>
> Regards
>
> Alan Powell
> Pigeon slayer
>
> At 03:07 AM 6/30/01, you wrote:
> >take a prime number (p), raise it to any power (n),
> >make a rectangle out of that number (with sides
> >p^(n-k) and p^k). Calculate the perimeter of that
> >rectangle. Add or subtract 1. At least one of these
> >will be a prime number ... I think. At any rate,
> all
> >the primes may be generated this way ... I think.
>
>

=====
Bill Krys
Email: billkrys@...

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• ... at random: n=11, k=3: 2*(2^8+2^3)+1 trivially factors as: 23^2 2*(2^8+2^3)-1 trivially factors as: 17*31 with zillions more where those came from...
Message 3 of 9 , Jun 30, 2001
> can you find any for me that don't work for
> e=2(2^(n-k)+2^k)?
at random: n=11, k=3:
2*(2^8+2^3)+1 trivially factors as: 23^2
2*(2^8+2^3)-1 trivially factors as: 17*31
with zillions more where those came from...
Sorry Bill, but the prime cicle is not so easily
rectangulated:-)
David
• Okay, Dave, but what about the other half of the conjecture that all primes may be generated from 1 in such manner, despite creating many composites? Can you
Message 4 of 9 , Jul 1, 2001
Okay, Dave,

but what about the other half of the conjecture that
all primes may be generated from 1 in such manner,
despite creating many composites? Can you find a
counter example?

Bill

=====
Bill Krys
Email: billkrys@...

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• ... Also false, I believe. Consider the prime q=103. I can see no way of writing either 51 or 52 in the form p^a+p^b. David (not Dave:-)
Message 5 of 9 , Jul 1, 2001
Bill Krys wrote:

> but what about the other half of the conjecture that
> all primes may be generated

Also false, I believe. Consider the prime q=103.
I can see no way of writing either 51 or 52
in the form p^a+p^b.

David (not Dave:-)
• To: Alan Powell Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent:
Message 6 of 9 , Jul 1, 2001
To: Alan Powell <powella@...>
From: Bill Krys <billkrys@...>
Date sent: Sat, 30 Jun 2001 06:16:28 -0700 (PDT)

> Dear Alan et al,
>
> can you find any for me that don't work for
> e=2(2^(n-k)+2^k)?

2*(2^(8-1)+2^1)-1 factor : 7
2*(2^(8-1)+2^1)+1 factor : 3
2*(2^(8-4)+2^4)-1 factor : 3
2*(2^(8-4)+2^4)+1 factor : 5
2*(2^(9-1)+2^1)-1 factor : 5
2*(2^(9-1)+2^1)+1 factor : 11
2*(2^(9-3)+2^3)-1 factor : 11
2*(2^(9-3)+2^3)+1 factor : 5
2*(2^(10-1)+2^1)-1 factor : 13
2*(2^(10-1)+2^1)+1 factor : 3
2*(2^(10-4)+2^4)-1 factor : 3
2*(2^(10-4)+2^4)+1 factor : 7

>
> > At 03:07 AM 6/30/01, you wrote:
> > >take a prime number (p), raise it to any power (n),
> > >make a rectangle out of that number (with sides
> > >p^(n-k) and p^k). Calculate the perimeter of that
> > >rectangle. Add or subtract 1. At least one of these
> > >will be a prime number ... I think. At any rate,
> > all
> > >the primes may be generated this way ... I think.
> >
> >
>
>
> =====
> Bill Krys
> Email: billkrys@...
> Toronto, Canada (currently: Beijing, China)
>
> __________________________________________________
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> Get personalized email addresses from Yahoo! Mail
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> The Prime Pages : http://www.primepages.org
>
>
>
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>
>

Michael Hartley : Michael.Hartley@...
Sepang Institute of Technology
+---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
"If you entertain a thought, it becomes an attitude..."
• To: d.broadhurst@open.ac.uk Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent: Sun, 1
Message 7 of 9 , Jul 1, 2001
From: Bill Krys <billkrys@...>
Date sent: Sun, 1 Jul 2001 08:33:19 -0700 (PDT)
Subject: Re: [PrimeNumbers] Re: sieve = 2(p^(n-k)+p^k)

> Okay, Dave,
>
> but what about the other half of the conjecture that
> all primes may be generated from 1 in such manner,
> despite creating many composites? Can you find a
> counter example?

Yes. 103 cannot be written 2(p^k + p^(n-k))+/-1 for any prime p,
integers n >= k >= 0.

Proof: Left as an exercise to the reader.

Puzzle: Find the _next_ such prime...

[ PS - I almost embarrassed myself by saying "Yes. 31 cannot..." ]

>
> Bill
>
> =====
> Bill Krys
> Email: billkrys@...
> Toronto, Canada (currently: Beijing, China)
>
> __________________________________________________
> Do You Yahoo!?
> Get personalized email addresses from Yahoo! Mail
> http://personal.mail.yahoo.com/
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

Michael Hartley : Michael.Hartley@...
Sepang Institute of Technology
+---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
"If you entertain an attitude, it becomes an action..."
• ... 103,139,151,157,199,223,233,239,241,307,311,313,353,367,373,379, 409,419,421,431,433,439,443,463,571,593,599,601,607,619,631,643,
Message 8 of 9 , Jul 1, 2001
> Puzzle: Find the _next_ such prime...
103,139,151,157,199,223,233,239,241,307,311,313,353,367,373,379,
409,419,421,431,433,439,443,463,571,593,599,601,607,619,631,643,
659,661,673,683,727,733,739,743,751,757,809,811,823,827,829,853,
857,859,877,883,911,919,941,947,953,967,991,997...
• Next puzzle: Why is EIS sequence ID Number: A033227 Sequence: 43,103*,139*,157*,181,277,367*,439*,523,547, 607*,673*,751*,823*,991*,997*... Name: Primes
Message 9 of 9 , Jul 1, 2001
Next puzzle: Why is EIS sequence

ID Number: A033227
Sequence: 43,103*,139*,157*,181,277,367*,439*,523,547,
607*,673*,751*,823*,991*,997*...
Name: Primes of form x^2+39*y^2.

such a fecund source of exceptions to Bill's conjecture?

[*] 11 of the first 16 elements are non-Krys
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