## Re: [PrimeNumbers] Re: sieve = 2(p^(n-k)+p^k)

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• To: d.broadhurst@open.ac.uk Copies to: primenumbers@yahoogroups.com From: Bill Krys Date sent: Sun, 1
Message 1 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)
>
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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 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 2 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 3 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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