a primesieve?

[bhelmes]

A peaceful evening for all,

Is it possible to find a prime generating sieving algorithm
for the following described function:
http://devalco.de/poly_xy.php

Have a lot of fun
Bernhard

[science_man_88]

Originally Posted by bhelmes

A peaceful evening for all,

Is it possible to find a prime generating sieving algorithm
for the following described function:
http://devalco.de/poly_xy.php

Have a lot of fun
Bernhard

do your own work ... also a quick PARI/GP script shows these are the u,v pairs mod 6 that work to give a possible prime greater than 3.

0,1
0,5
1,0
1,2
1,4
2,1
2,3
2,5
3,2
3,4
4,1
4,3
4,5
5,0
5,2
5,4

[bhelmes]

A peaceful day for all,

Originally Posted by science_man_88

do your own work ... also a quick PARI/GP script shows these are the u,v pairs mod 6 that work to give a possible prime greater than 3.

The described primes are also p+/-1=0 mod 8

Hint 1: Fix either u or v for the function f(u,v) and examine the prime distribution. There is a pattern, you could find.

http://devalco.de/poly_xy.php

Have a lot of fun by examing the prime distribution
Bernhard

[bhelmes]

Originally Posted by bhelmes

A peaceful day for all,

Hint 1: Fix either u or v for the function f(u,v) and examine the prime distribution. There is a pattern, you could find.

Hint 2: Consider for f(u,v)=f(u, u-1), there is an explication for a primesieve concerning f(n)=2n^2-1 : http://devalco.de/quadr_Sieb_2x%5E2-1.php

You can use this primesieve for a better primesieve concerning
http://devalco.de/poly_xy.php

There will be a third hint in some few days, and concerning the participation a suggestion for a first solution.

There might be several solution, you have the first choice
Greeting from the primes
Bernhard

[science_man_88]

Originally Posted by bhelmes

A peaceful day for all,



The described primes are also p+/-1=0 mod 8

Hint 1: Fix either u or v for the function f(u,v) and examine the prime distribution. There is a pattern, you could find.

http://devalco.de/poly_xy.php

Have a lot of fun by examing the prime distribution
Bernhard

if there's a pattern you know about then just do it, and stop posting it where I have to look at it then ?? you seem too much like me.

[Batalov]

This looks like Misc.Math.
(it is obviously not a puzzle, just a bunch of fluff plus shameless self-promotion to boot.)

[bhelmes]

"Anyone who has "discovered" a pattern or found order in disorder,ie. feigenbaum constants, are justified in feeling
a sense of pride and accomplishment for this personal insight. " from jwaltos (http://www.mersenneforum.org/showpos...9&postcount=25)

I am very thankful for these words.

An explication for the proposed pattern for the function
f(u,v)=u^2 + 2uv - v2 :

You can simplify by fixing either u or v and you get a quadratic function with one variable. You could use this "slices" horizontal and vertikal.
for every p with p | f(n)=an^2+bn+c (a<5; a,b,c element N) follows that
p | f(n+p) and p | f(-n-b/a)
The primes p with p | f(u,v) appears double periodically horizantal and vertikal.

In order to understand this i have tried to give an "amazing" applet
http://devalco.de/poly_xy.php
you can click on the primes 7, 17, 23, 31 and so on

The primes p of the function f(u,v)=u^2 + 2uv - v2 build corally a pattern.

The remaining question is:
in which order you can make a sieving (dividing by the primes)

As the pattern is clear, the solution for the algorithm will come in 3 days.
Bernhard

@Batalov
I support Gimps, LL-D test place 164

[CRGreathouse]

Originally Posted by bhelmes

An explication for the proposed pattern for the function
f(u,v)=u^2 + 2uv - v2

If I understand you correctly, this is A038873, 2 together with primes ±1 mod 8.

[bhelmes]

Originally Posted by CRGreathouse

If I understand you correctly, this is A038873, 2 together with primes ±1 mod 8.

Right

[bhelmes]

A peaceful evening for all,

a proposition for a primesieve concerning f(u, v)=u² + 2uv -v² :

1. Make a primesieve concerning the function f(u, u-1) = 2u²-1
from u=2 to u_max (there is a detailled description under
http://devalco.de/quadr_Sieb_2x%5E2-1.php

Safe all sieved out primes p with p < u_max,
these primes appear in all quadratic functions with f(u, v)
where u goes from 2 to u_max.
(This should be the amount Q)

2. Make a sieving (vertikal) for f(u, 1)= u²+2u-1 by using the presieved primes Q

3. Make a sieving (vertikal) for f(u, 2)= u²+4u-4 by using the presieved primes Q

4. Make a sieving (vertikal) for f(u, 3)= u²+6u-9 by using the presieved primes Q

....

u-2. Make a sieving (vertikal) for f(u, u-2)= u²+2(u-1)-(u-2)² by using the presieved primes Q

The sieving is made by dividing the function f(u,v) by the primes Q.

Do you have any idea why this prime sieve could be better than the prime generator for the function f(n)=2n²-1 ?

@Batalov
It looks like Misc. Math is a cold, dark and windy place,
do you have a more decent place for the thread ?
Or please give a mathematical reason.