Prime Numbers

106 Chapter 2 NUMBER-THEORETICAL TOOLS
2. [Function newr() that gives ri+1 from ri]
newr(ri) {
x = f(ri)p−2i; z =(f ′ (r)) −1 mod p2i; // Via Algorithm 2.1.4.
y = −xz mod p2i; ri+1 = ri + yp2i; return ri+1;
}
2 Note that for j ≥ i we have rj ≡ ri mod p i
, so that the sequence (ri)
converges in the p-adic numbers to a root of f(x). In fact, Hensel lifting may
be regarded as a p-adic version of the Newton methods discussed in Section
9.2.2.
2.3.4 Representation by quadratic forms
We next turn to a problem important to such applications as elliptic curves
and primality testing. This is the problem of finding quadratic Diophantine
representations, for positive integer d and odd prime p, in the form
x 2 + dy 2 = p,
or, in studies of complex quadratic orders of discriminant D