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

To construct quadratics,

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