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Gerry
science forum beginner

Joined: 11 Nov 2005
Posts: 23

Posted: Tue Jul 18, 2006 10:45 am    Post subject: Primality & Factoring

Two questions relating the following function:

N(r,x,y,z)=r^2+(z^2y+(2z+1)x)r+z^2(z^2-1)y^2/4+z(z^2+(z-1)/2)yx+z(z+1)x^2

1) How can i determine solutions for variables r,x,y,z if N is given?

2) For which range of the variables r,x,y,z can N be prime?

Gerry
science forum beginner

Joined: 11 Nov 2005
Posts: 23

 Posted: Tue Jul 18, 2006 12:57 pm    Post subject: Re: Primality & Factoring For example 493=N(r,x,y,z)=N(5,4,4,2)=N(9,8,4,1)=N(11,6,6,1)=N(17,12,8,0) ( Variables r,x,y,z >0 and integer are of interest)
Dave Rusin
science forum Guru

Joined: 25 Mar 2005
Posts: 487

Posted: Tue Jul 18, 2006 5:26 pm    Post subject: Re: Primality & Factoring

Gerry <GerryMrt@gmail.com> wrote:

 Quote: Two questions relating the following function: N(r,x,y,z)=r^2+(z^2y+(2z+1)x)r+z^2(z^2-1)y^2/4+z(z^2+(z-1)/2)yx+z(z+1)x^2

i.e. 4N = (2r + z^2y+2xz+x )^2 -(x + yz)^2

 Quote: 1) How can i determine solutions for variables r,x,y,z if N is given?

Factor N = pq, so 4N = u^2 - v^2 with u = p+q, v = p-q. Then for
any y,z let x = p - q - yz, r = (p+q - (x+2xz+yz^2))/2 (which
turns out to be integral).

 Quote: 2) For which range of the variables r,x,y,z can N be prime?

This requires q to be in {1, -1, N, -N}. For each fixed q the "range"
is parameterized by the (y,z) pairs as above.

Wrong newsgroup. Try sci.math.

dave
Gerry
science forum beginner

Joined: 11 Nov 2005
Posts: 23

 Posted: Thu Jul 20, 2006 4:12 pm    Post subject: Re: Primality & Factoring 1) Interesting approach. Would it be possible to create a faster algorithm than normal factoring by trying to fit p and q as follows: N=pq, p=yz^2/2+(x-y/2)z+r , q=y(z+1)^2/2+(x-y/2)(z+1)+r Gerry Sorry about the wrong group.

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