Re: [PrimeNumbers] Web--of--Ones
- Wednesday, March 15, 2006 8:11 AM [GMT+1=CET],
Phil Carmody <thefatphil@...> escribió:
> --- aldrich617 <aldrich617@...> wrote:Thanks to Phil and Jose, I was slightly obtuse ...
>> I am searching for a way to prove the following Theorem:
>> For any integer 'B', the value 'A' of the equation
>> A = 5B^4 -10B^3 + 20B^2 -15B +11 will have as factors only
>> integers that end in a one, excluding all others.
>> This seems to be true at least up to 10^18. I think that proving
>> it could give us new insights into primality testing,
>> and factoring. Moreover there are similar equations, vast in
>> number, apparently with the same property, that could then probably
>> be verified to be similar threads of pure one. Together these
>> would form an infinite interconnecting web.
> Can you tell us how you discovered that polynomial?
> The discriminant is very smooth, being 5^3*11^2, and I suspect that
> that's an essential ingredient to a proof.
> It seems that if p== +/-1 mod 10, then your polynomial splits at
> least into 2 quadratics, and if p== +1 mod 10, then at least one of
> those quadratics splits.
> This property should be easily explainable, but alas it's late and my
> brain's on holiday. (I wrote this last night, but forgot to send).
But, someone could tell us how to prove that it must be p = 1 (mod 10)?
Thanks in advance,
Ignacio Larrosa Cañestro
A Coruña (España)