Elliptic curves in NFS
Date: 17 Nov 2005 06:02:30 -0800
Here is a probably useless relation between elliptic curves and NFS.
Consider a NFS with polynomial of degree 2.
The product of the rational and the algebraic sides is of the form:
Consider the elliptic curve
If one knows a rational point on this elliptic curve, by multiplying
rational (x,y) may be generated.
The case for Fermat numbers seems interesting (PARI session):
For example F5:
po=(x^2+1)*(x+ii) == x^3 + 65536*x^2 + x + 65536
the point1=[0,s1] is on the curve.
%30 = 1
so both the norm and the rational side are squares and in addition y1
(all in Z).
Is there an analytical explanation that these "squares" will lead to
*trivial* results - because the squares seem the final stage of NFS
sieving skipped in the case of Fermat numbers ?
\\ PARI code