- A slight modification in this polynomial yields a slightly higher

count of primes. In searching for primes of the form ax^2+bx+c, it

is good to put the axis of symmetery at -b/2a=50000.5 so that when x

yields a prime value then 100001-x also yields a prime value (what we

Americans call a "two-fer," for two for one sales). The general

principle is sound but one should realize that, once a candidate

formula is found, the value at x=1 might be composite while the value

at x=100001 might be prime, so that shifting x by 1 would up the

prime count. In this fashion I identified a shift of 5357 giving 16

more primes. The polynomial x^2-89287*x+1991687729 gives 44500 Prp

on the range x=1..100000.

Adam Stinchcombe

snip>>>>

Try to beat my record in CYF NO. 13 at

www.shyamsundergupta.com/canyoufind.htm> This allows negative and repeated primes and I got 44484 primes to

100000 in

> x^2-100001x+2498695637

a prime

> The puzzle asks for 50000 but that is beyond me.

> According to my computations, 92.4% of the candidates will not have

> factor below 250 for x^2+x-7731189253. It only gave 41503 primes to

100000.

<<<snip - Adam wrote:
> A slight modification in this polynomial yields a slightly higher

Nice work. I also considered trying to shift when I found the symmetric

> count of primes. In searching for primes of the form ax^2+bx+c, it

> is good to put the axis of symmetery at -b/2a= so that when x

> yields a prime value then 100001-x also yields a prime value (what we

> Americans call a "two-fer," for two for one sales). The general

> principle is sound but one should realize that, once a candidate

> formula is found, the value at x=1 might be composite while the value

> at x=100001 might be prime, so that shifting x by 1 would up the

> prime count. In this fashion I identified a shift of 5357 giving 16

> more primes. The polynomial x^2-89287*x+1991687729 gives 44500 Prp

> on the range x=1..100000.

>

> Adam Stinchcombe

quadratic. I knew the requested 50000 primes was far out of reach and was too

lazy to try the shift. I did no heuristics but did not expect as many as 16

extra primes.

Brian Trial's best attempt at www.shyamsundergupta.com/canyoufind.htm is a

shift of my second best, but he found it independently. It only has one extra

prime and only shifts the symmetri axis from 50000.5 to 49999.5. Maybe he did

not shift but started with a slightly asymmetric quadratic on the interval

1..100000.

--

Jens Kruse Andersen