## Re: RE 40 prime polynomials

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• ... A modification is in order. Any prime sequence which is found in ax^2 + sx + t where s =2a can be found in the simpler ax^2 + bx + c where b
Message 1 of 12 , Apr 5, 2006
<mark.underwood@...> wrote:
>
>
> Put differently, any sequence which is found in
> ax^2 + sx + t
> where s >= 2a,
> can be found in the simpler
> ax^2 + bx + c
> where b < 2a.
>

A modification is in order. Any prime sequence which is found in
ax^2 + sx + t where s>=2a
can be found in the simpler
ax^2 + bx + c
where b <= a and both a and b are positive.

For instance, consider 5 polys which generate 40 consecutive primes
from x=0 to x=39.

x^2 + x + 41
4x^2 - 154x + 1523
4x^2 - 158x + 1601
9x^2 - 231x + 1523
9x^2 - 471x + 1603

The second and third polys generate the same sequence (in reverse),
and so does the fourth and fifth polys.

So really there are three:
x^2 + x + 41
4x^2 - 154x + 1523
9x^2 - 231 + 1523.

But these three can be simplified to
x^2 + x + 41 (prime from x=0 to x=39)
4x^2 + 2x + 41 (prime from x=-20 to x=19)
9x^2 + 3x + 41 (prime from x=-20 to x=19)

Here's another 40 I stumbled upon, but which dips into the negatives
in the middle of the sequence:

8x^2 + 6x - 661 (prime from x=-19 to x=20)

Anyways, the point I wanted to reiterate is that if one is searching
for prime polys of the form
ax^2 + bx + c
such that x roams in a certain range, one need only consider positive
b's from 0 to a. (Otherwise there is redundancy.)

Mark
• ... Another example: Consider the (already known) prime poly 36x^2 - 810x + 2753 which produces 45 consecutive primes (including negatives) from x=0 to x=44.
Message 2 of 12 , Apr 5, 2006
<mark.underwood@...> wrote:
>
>
> Anyways, the point I wanted to reiterate is that if one is searching
> for prime polys of the form
> ax^2 + bx + c
> such that x roams in a certain range, one need only consider positive
> b's from 0 to a. (Otherwise there is redundancy.)
>

Another example: Consider the (already known) prime poly
36x^2 - 810x + 2753
which produces 45 consecutive primes (including negatives) from x=0 to
x=44.

This poly can be reduced to
36x^2 + 18x - 1801
which is prime from x=-33 to x=11.
Notice how b is positive and no more than a. Also notice how b contains
all the prime factors of a. It must, or else the poly is forced to
contain the prime factors of a which b doesn't have.

Mark
• ... wrote: ... Correction: 9x^2 + 3x + 41 is prime from x=-13 to x=26 Mark (no alcohol required to goof up) Underwood
Message 3 of 12 , Apr 5, 2006