## Improving Jens Kruse Andersen quadratic primes

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• 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
Message 1 of 16 , May 4, 2003
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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.

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
> The puzzle asks for 50000 but that is beyond me.
> According to my computations, 92.4% of the candidates will not have
a prime
> factor below 250 for x^2+x-7731189253. It only gave 41503 primes to
100000.
<<<snip
• ... Nice work. I also considered trying to shift when I found the symmetric quadratic. I knew the requested 50000 primes was far out of reach and was too lazy
Message 2 of 16 , May 5, 2003
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> 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= 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.
>

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