Prime Numbers

176 Chapter 4 PRIMALITY PROVING
smaller than F . From the uniqueness of the base-F representation of a number
it follows that c2 = ab and c1 = a + b as claimed.
Now suppose, conversely, that c 2 1 − 4c2 is a square, say u 2 .Then
c1 + u c1 − u
n = F +1 F +1 .
2 2
The two fractions are both integers, since c1 ≡ u (mod 2). It remains to note
that this factorization is nontrivial, since c2 > 0 implies |u|