Prime Numbers

188 Chapter 4 PRIMALITY PROVING
if so, xai + ybi will be one of the numbers c computed in Step [Loop] of
Algorithm 4.2.11, because of (4.20). Thus, Step [Loop] will successfully retrieve
the numbers x, y.
We have xa0 + yb0 = xa0 ≥ 0andxat + ybt = ybt ≤ 0, so there is some
even index i with
xai + ybi ≥ 0, xai+2 + ybi+2 ≤ 0.
If one of these quantities is less than s in absolute value, we are done, so
assume that the first quantity is ≥ s and the second is ≤−s. Thenfrom
(4.17), (4.18), (4.19),
xai ≥ xai + ybi ≥ s = bi+1ai − ai+1bi ≥ bi+1ai,
from which we conclude that x ≥ bi+1. Wealsohave
ybi+2 ≤ xai+2 + ybi+2 ≤−s = bi+2ai+1 − ai+2bi+1 ai+1. Therefore,
xai+1 + ybi+1 > 2ai+1bi+1,
and from (x − bi+1)(y − ai+1) > 0, we have
xai+1 + ybi+1 ≤ xy + ai+1bi+1