Prime Numbers

186 Chapter 4 PRIMALITY PROVING
polynomial vx 3 − (uF − c1v)x 2 − (c4v − dF + u)x + d has no integral
root a such that aF +1 is a nontrivial factor of n, and the polynomial
vx 3 +(uF − c1v)x 2 − (c4v + dF + u)x + d has no integral root b such
that bF − 1 is a nontrivial factor of n.
The next result allows one to combine partial factorizations of both n − 1
and n + 1 in attempting to prove n prime.
Theorem 4.2.10 (Brillhart, Lehmer, and Selfridge). Suppose that n is a
positive integer, F1|n−1, and that (4.3) holds for some integer a1 and F = F1.
Suppose, too, that f,∆ are as in (4.12), gcd(n, 2b) =1,
∆
n = −1, F2|n +1,
and that (4.14) holds for F = F2. LetFbe the least common multiple of
F1,F2. Then each prime factor of n is congruent to either 1 or n (mod F ).
In particular, if F> √ n and n mod F is not a nontrivial factor of n, then n
is prime.
Note that if F1,F2 are both even, then F = 1
2 F1F2, otherwise F = F1F2.
Proof. Let p be a prime factor of n. Theorem 4.1.3 implies p ≡ 1(modF1),
while Theorem 4.2.3 implies that p ≡
∆
p (mod F2). If
∆
p =1,thenp≡1 (mod F ), and if
∆
p = −1, then p ≡ n (mod F ). The last assertion of the
theorem is then immediate. ✷
4.2.3 Divisors in residue classes
What if in Theorem 4.2.10 we have F