## 24107Re: Two large consecutive smooth numbers

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• Mar 4, 2012
On 3/4/2012 7:59 AM, primenumbers@yahoogroups.com wrote:
> 1c. Re: Two large consecutive smooth numbers
> Posted by: "Phil Carmody"thefatphil@... thefatphil
> Date: Sat Mar 3, 2012 3:03 pm ((PST))
>
> Calling Doctor Broadhurst for suggestion of the best metric by which to evaluate such records. A simple log doesn't necessarily tell the whole tale at all.
>
> Be warned, Warren - Dr. B is sitting on a corpus of algebraic formulae such that p(x) and p(x)+1 have algebraic factorisations, which makes smoothness measurably (I was going to say immeasurably, and then realised the stupidity of such a word choice) more likely.
>
> I'll not play this game, as I have an appointment with 21 farmers in Lithuania (otherwise known as the biggest brewery crawl yet...)
>
> Phil

x^2 - b x + c and x^2 - b x + c + 1 which are both factorisable,

we look for integers b, k1, k2 such that

c = k1 * (b-k1)
and
c+1 = k2 * (b-k2)

1*3 = 3; 2 * 2 = 4

x^2 - 4 x + 3 = (x-1)*(x-3)
x^2 - 4 x + 4 = (x-2)^2

2 * 4 = 8; 3 * 3 = 9

x^2 - 6 x + 8 = (x-2)*(x-4)
x^2 - 6 x + 9 = (x-3)^2 which is really the same as the first example
translated by 1.

I will guess that maybe the only quadratic polynomial solutions are
translations of the first example.

Cubic polynomial solutions should be more prolific.

Kermit
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