Prime Numbers

228 Chapter 5 EXPONENTIAL FACTORING ALGORITHMS
It remains to show that k = uv ≤ n1/3 .Since u
v
have
k = uv = u
v v2 < p
q v2 + v p q
≤ ·
B q p n1/3 +1=n 1/3 +1,
p 1 < q + vB
and v ≤ B, we
so the claim is proved.
With k, u, v as above, let a = uq + vp, b = |uq − vp|. Then4kn = a2 − b2 .
We show that 2 √ kn ≤ a < 2 √ kn + n1/6
4 √ . Since uq · vp = kn, we have
k
a = uq + vp ≥ 2 √ kn. Seta =2 √ kn + E. Then
4kn +4E √ kn ≤
2 √ 2 kn + E = a 2 =4kn + b 2 < 4kn + n 2/3 ,
so that 4E √ kn < n2/3 ,andE< n1/6
4 √ as claimed.
k
Finally, we show that if a, b are returned in Step [Loop], then gcd(a + b, n)
is a nontrivial factor of n. Sincendivides (a + b)(a − b), it suffices to show
that a + b