Prime Numbers

184 Chapter 4 PRIMALITY PROVING
(mod (f(x),M)) and by the Euler criterion we have 2 (M−1)/2 ≡ (2/M )=1
(mod M), so
Next,
(x − 1) M+1 ≡ (2x) (M+1)/2 =2· 2 (M−1)/2 x (M+1)/2
≡ 2x (M+1)/2 (mod (f(x),M)).
(x − 1) M+1 =(x − 1)(x − 1) M ≡ (x − 1)(x M − 1) ≡ (x − 1)(3 − x)
≡−2(mod(f(x),M)).
Thus, x (M+1)/2 ≡−1(mod(f(x),M)); that is, x2p−1 ≡−1(mod(f(x),M)).
Using our automorphism, we also have (4 − x) 2p−1 ≡−1(mod(f(x),M)), so
that U2p−1 ≡ 0(modM). If U2p−2 ≡ 0(modM), then x2p−2 ≡ (4 − x) 2p−2
(mod (f(x),M)), so that
−1 ≡ x 2p−1
≡ x 2p−2
(4−x) 2p−2
≡ (x(4−x)) 2p−2
≡ 1 2p−2
≡ 1(mod(f(x),M)),
a contradiction. Since U 2 p−1 = U 2 p−2V 2 p−2, wehaveV 2 p−2 ≡ 0(modM). But
we have seen that V 2 p−2 = vp−2, so the proof is complete. ✷
Algorithm 4.2.7 (Lucas–Lehmer test for Mersenne primes). We are given
an odd prime p. This algorithm decides whether 2 p −1 is prime (YES) or composite
(NO).
1. [Initialize]
v =4;
2. [Compute Lucas–Lehmer sequence]
for(k ∈ [1,p− 2]) v =(v 2 − 2) mod (2 p − 1); // k is a dummy counter.
3. [Check residue]
if(v == 0) return YES; // 2 p − 1 definitely prime.
return NO; // 2 p − 1 definitely composite.
The celebrated Lucas–Lehmer test for Mersenne primes has achieved some
notable successes, as mentioned in Chapter 1 and in the discussion surrounding
Algorithm 9.5.19. Not only is the test breathtakingly simple, there are ways
to perform with high efficiency the p−2 repeated squarings in Step [Compute
Lucas–Lehmer sequence].
4.2.2 An improved n +1 test, and a combined n 2 − 1 test
As with the n − 1 test, which is useful only in the case that we have a large,
fully factored divisor of n − 1, the principal hurdle in implementing the n +1
test for most numbers is coming up with a large, fully factored divisor of
n + 1. In this section we shall improve Theorem 4.2.3 to get a result similar
to Theorem 4.1.5. That is, we shall only require the fully factored divisor of
n + 1 to exceed the cube root. (Using the ideas in Theorem 4.1.6, this can be
improved to the 3/10 root.) Then we shall show how fully factored divisors