- Here's one that won't be a winner but is fairly long. I can't find it

in the Integer Sequences, so maybe it's 'new':

4x^2 - 212x + 2411

Generates 27 distinct primes from x=0 (p=2411) to x=26 (p=-397)

before doubling back. So it's prime from x=0 to x=53. It met its end

at the hands of a factor of 37.

Mark

--- In primenumbers@yahoogroups.com, ed pegg <ed@m...> wrote:

>

> The next Al Zimmermann Programming contest might be about

> Prime Generating Polynomials. It's currently running neck

> and neck with Protein Folding.

> http://www.recmath.com/contest/votes.php

>

> Under the rules I'm proposing for the contest, here are the

> current known record holders for Prime Generating Polynomials,

> orders 1 to 4. Orders 5 and up seem to be unexplored.

>

> 1) 44546738095860 n + 56211383760397 Score:23 n=0..22 Frind

> 2) 36 n^2 - 810 n + 2753 Score:45 n=0..44 Ruby

> 3) 3 n^3 - 183 n^2 + 3318 n - 18757 Score:43 n=0..46 Ruiz

> 4) n^4 + 29 n^2 + 101 Score:20 n=0..19 Pegg

>

> The scoring rules:

> 1. Polynomial f(k) must produce primes from 0 to n.

> 2. The score will be the number of *distinct* primes

> when |f(k)| is evaluated from 0 to n.

> 3. In case of a tie, the lower value (tighter value) of n wins.

> 4. In case of a tie, the product of non-zero coefficients will

> be evaluated, and the lowest product wins.

>

> My own result for order-4 polynomials will likely be surpassed

> very easily. If you would like to vote for this contest, please

> visit http://www.recmath.com/contest/ . The winners will split

> up $500.

>

> Ed Pegg Jr

> - Dick wrote:

> A quadratic expression that yields no prime factors less than a given

Yes. Here is PARI/GP code:

> value is easy to generate arbitrarily.

f(n)=local(c,p,i,r);c=1;r=2;forprime(p=3,n,i=1;

while(issquare(Mod(i,p)),i++);c+=lift((Mod(1-i,p)/4-c)/r)*r;r*=p);c

x^2+x+f(n) never has a factor <= n.

f(n) is a little below n# (which is near e^n).

f(277) has 113 digits:

52211040781253690937101509813868236062339335365181118768\

264647548098518281973426206257327260311399656484832821211

f(100000) is computed in 2 seconds and has 43293 digits.

It is far harder to find the smallest constants to avoid

all small factors in x^2+x+A.

http://www.primepuzzles.net/conjectures/conj_017.htm lists that.

The record is A=2457080965043150051 which gives 281 as smallest factor.

--

Jens Kruse Andersen