## Proth's theorem extended

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• Hello, let Q= k*2^n +1, k
Message 1 of 7 , Jul 28, 2008
Hello,
let Q= k*2^n +1, k< 2^n +1 be odd, and 'n' be prime.
Proth extended: if a^((Q-1)/4) == +/-1(mod Q), then 'Q' is prime.
my rule: also, if a= 2 & n is prime, then Q will NEVER be pseudo-prime.

extended portion:
if m/is a natural number/, then every odd divisor 'q' of a^(2^(m+1)) +1 implies that q == 1(mod(a^(m+2))) [concluded from generalized Fermat-number 'proofs' by Proth himself but replacing 'm' with (m+1) in his original argument]. similarly, a^(2^(m+1)) -1 implies q == -1(mod(a^(m+2))) by the same observations.

Now, if 'p' is any prime divisor of 'R', then a^((R-1)/4) = (a^k)*(a^(n-2)) == +/- 1(mod p) implies that either p == 1(mod(a^n)) or p == -1(mod (a^n)).

Thus, if 'R' is composite, 'R' will be the product of at least two primes each of which has minimum value 2^n +1; quickly...

2^n +1 > 0, 2^n > -1, k*2^n > -k, and k*2^n +1 > -k +1; so 2^n +1 would be the minimum choice instead of the factor... 2^n -1 > 0, 2^n > +1, k*2^n > +k, and k*2^n +1 > k +1 where both 'k's are odd; 2^n +1 would be minimal.

k*2^n +1 >= (2^n +1) * (2^n +1) = (2^n)*(2^n) + 2*(2^n) +1; 1's cancel so,
k*(2^n) >= 2^n*(2^n) +2(2^n) and is k >= 2^n +2 is a contradiction. This was exactly how Proth found the boundary for 'k' except I had to rule out the other 'quickly...' supposed minimal factor choice.

Hence, if a^((Q-1)/4) == +/-1(mod Q), then 'Q' is also prime for k< 2^n +1.
*QED

my rule: looking at Q-1 = 2^(2s)*f; if n = 2, k = 1 and Q= 1*2^2 + 1, and ((Q-1)/4) == 1(mod Q) is trival; but if n is and odd prime then Q-1 = (2^(2s))*f where f = 2*g and g= 2^n+1; so dividing by 4 means that f = 2^(2s)*(2^n) and upon examination of any psuedo-prime minus one... I can easily conclude that Q-1 = (only a product of 2's) isn't the correct composition for a pseudo-prime minus one.

Hence, if a= 2 and n is prime, then Q will NEVER produce a pseudo-prime.

does anyone concur... ??? wjb
• I have extended Proth s theorem. It can be found on my website... www.oddperfectnumbers.com under the Other Short Proofs heading. You will also find the proof
Message 2 of 7 , Apr 9, 2011
I have extended Proth's theorem.

It can be found on my website...
www.oddperfectnumbers.com under

You will also find the proof for
"why odd-perfect numbers don't
exist" among others. Enjoy!

Bill
• Hi, I am afraid that your idea why odd-perfect numbers don t exist would not work due to the existence of odd abundant numbers. Best regards, Dimiter
Message 3 of 7 , Apr 11, 2011
Hi,

I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.

Best regards,
Dimiter

--- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
>
> I have extended Proth's theorem.
>
> It can be found on my website...
> www.oddperfectnumbers.com under
> the Other Short Proofs heading.
>
> You will also find the proof for
> "why odd-perfect numbers don't
> exist" among others. Enjoy!
>
> Bill
>
• Dimiter, first of all, you haven t even visited my website... and second, the argument is so simple that it does NOT require that someone even consider the
Message 4 of 7 , Apr 11, 2011
Dimiter,

first of all, you haven't even visited my website...
and second, the argument is so simple that it does NOT
require that someone even consider the summand portions
which would make the answer deficient or abundant. The
proof combines Euler's work w/that of Jacques Tochard's
from 1953; I just threaded the ideas together. (prime,
prime, prime; I have to mention them not to get spammed)
Bill

--- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@...> wrote:
>
> Hi,
>
> I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
>
> Best regards,
> Dimiter
>
> --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@> wrote:
> >
> > I have extended Proth's theorem.
> >
> > It can be found on my website...
> > www.oddperfectnumbers.com under
> > the Other Short Proofs heading.
> >
> > You will also find the proof for
> > "why odd-perfect numbers don't
> > exist" among others. Enjoy!
> >
> > Bill
> >
>
• Bill, please explain why solutions to: 2*P*Q^2 == (P+1)*sigma(Q^2), with P,Q odd relatively prime numbers 1 exist, but why none exist when P is prime.
Message 5 of 7 , Apr 11, 2011
Bill, please explain why solutions to:

2*P*Q^2 == (P+1)*sigma(Q^2),
with P,Q odd relatively prime numbers > 1

exist, but why none exist when P is prime.
Because if one exists when P is prime, that
would give an odd perfect number.

(To see existence, try P=22021, Q=3003.)

On 4/11/2011 1:43 PM, leavemsg1 wrote:
> Dimiter,
>
> first of all, you haven't even visited my website...
> and second, the argument is so simple that it does NOT
> require that someone even consider the summand portions
> which would make the answer deficient or abundant. The
> proof combines Euler's work w/that of Jacques Tochard's
> from 1953; I just threaded the ideas together. (prime,
> prime, prime; I have to mention them not to get spammed)
> Bill
>
>
> --- In primenumbers@yahoogroups.com, "Dimiter Skordev"<skordev@...> wrote:
>>
>> Hi,
>>
>> I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
>>
>> Best regards,
>> Dimiter
>>
>> --- In primenumbers@yahoogroups.com, "leavemsg1"<leavemsg1@> wrote:
>>>
>>> I have extended Proth's theorem.
>>>
>>> It can be found on my website...
>>> www.oddperfectnumbers.com under
>>> the Other Short Proofs heading.
>>>
>>> You will also find the proof for
>>> "why odd-perfect numbers don't
>>> exist" among others. Enjoy!
>>>
>>> Bill
>>>
>>
>
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
>
>
>
• Of course I visited Bill s website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the
Message 6 of 7 , Apr 11, 2011
Of course I visited Bill's website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the link "Odd Perfect Numbers Don't
Exist" (it turned out this is just the root page of the site). The opinion I expressed in my previous message is based on the content of the above-mentioned page.

--- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
>
> Dimiter,
>
> first of all, you haven't even visited my website...
> and second, the argument is so simple that it does NOT
> require that someone even consider the summand portions
> which would make the answer deficient or abundant. The
> proof combines Euler's work w/that of Jacques Tochard's
> from 1953; I just threaded the ideas together. (prime,
> prime, prime; I have to mention them not to get spammed)
> Bill
>
>
> --- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@> wrote:
> >
> > Hi,
> >
> > I am afraid that your idea why odd-perfect numbers don't exist would not work due to the existence of odd abundant numbers.
> >
> > Best regards,
> > Dimiter
> >
> > --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@> wrote:
> > >
> > > I have extended Proth's theorem.
> > >
> > > It can be found on my website...
> > > www.oddperfectnumbers.com under
> > > the Other Short Proofs heading.
> > >
> > > You will also find the proof for
> > > "why odd-perfect numbers don't
> > > exist" among others. Enjoy!
> > >
> > > Bill
> > >
> >
>
• I m am sorry for accusing you of not visiting my website. I checked my Google Analytics program on the day of your visit , and it reported that no visits came
Message 7 of 7 , Apr 13, 2011
I'm am sorry for accusing you of not visiting my website.
I checked my Google Analytics program on the day of your
"visit", and it reported that no visits came in on that
day; I stand corrected! www.oddperfectnumbers.com

Bill

--- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@...> wrote:
>
> Of course I visited Bill's website, but I thought the author gives his explanations about the non-existence of odd perfect numbers in the target page of the link "Odd Perfect Numbers Don't
> Exist" (it turned out this is just the root page of the site). The opinion I expressed in my previous message is based on the content of the above-mentioned page.

> > It can be found on my website...
> > www.oddperfectnumbers.com under
> > the Other Short Proofs heading.

> > You will also find the proof for
> > "why odd-perfect numbers don't
> > exist" among others. Enjoy!
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