----- Original Message ----

From: Mark Underwood <mark.underwood@...>

To: primenumbers@yahoogroups.com

Sent: Monday, June 2, 2008 5:53:28 AM

Subject: [PrimeNumbers] Re: CoNtEsT uPdAtE !!!

--- In primenumbers@ yahoogroups. com, "aldrich617" <aldrich617@ ...> wrote:

>

>

> We apparently have a winner - the eminent

> Chris Caldwell of UTM. His opus delectus -

> a hideously beautiful ten page polynomial -

> is a masterwork of grace and economy of

> style. Though I am not personally qualified

> to see if his numbers really add up, I am

> inclined to simply take him on his word,

> because I know that sooner or later,

> somebody like DB (when he returns from Krypton)

> will zorch him good for any errors.

> He is hereby awarded the $91.80 Prize along

> with many thanks, good wishes and

> Undying Glory.

>

> The contest however will remain open, and I

> will smash my backup piggybank to replenish

> the prize, because what I really wanted was

> a tidy polynomial in one variable to send

> home to mum. I want a polynomial of less

> than ten degrees, preferably seven, of the form

> ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h,

> or something similar, and unless I get one

> I'm going to start claiming that my little

> procedure is really more succinct than anything

> the Math Wizards Society can concoct.

>

> a.

>

Aldrich, I have to say that your description of Prof. Caldwell's

solution - "opus delectus" and "hideously beautiful" - had me

laughing. Helped to cheer my day, thank you.

Like you, I have little idea about how Mr. Caldwell did what he did.

But I'm pretty sure that, if you retain the order of the numbers as

you presented them, you won't be able to generate those numbers with a

(one variable) polynomial of any order under about 90.

Why? For instance, look at the first 9 numbers you started with:

3167 2753 2753 1979 2357 1277 1979 647 1619.

I remember drawing smooth line graphs a very long time ago in grade

twelve (?) calculus, and doing the minima and maxima thing. Derivative

(or slope) equals zero at the minima and maxima. Generally speaking,

and if I recall correctly, a polynomial of degree two had one minima

or maxima (valleys or peaks), a poly of degree three had two valleys

or peaks, and so on. If you were to graph just the nine numbers above

with a smooth line you would see a very curvy line with seven peaks

and valleys. So, these first nine numbers alone would require a

polynominal of at least degree eight if I'm not mistaken.

So I think your "little procedure" for prime generation from a given

set of seed primes is proving much easier and interesting and fruitful

than the polynomial route. (Not that I inspected your procedure very

closely.) Mum would be proud. :)

Mark

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