Prime Numbers

5.1 Squares 227
5.1.2 Lehman method
But how do we know to try the multiplier 3 in the above example? The
following method of R. Lehman formalizes the search for a multiplier.
Algorithm 5.1.2 (Lehman method). We are given an integer n>21. This
algorithm either provides a nontrivial factor of n or proves n prime.
1. [Trial division]
Check whether n has a nontrivial divisor d ≤ n 1/3 , and if so, return d;
2. [Loop]
for 1 ≤ k ≤ n 1/3 {
for ⌈2 √ kn⌉ ≤a ≤⌊2 √ kn + n 1/6 /(4 √ k)⌋ {
if b = √ a 2 − 4kn is an integer return gcd(a + b, n);
// Via Algorithm 9.2.11.
}
}
return “n is prime”;
Assuming that this algorithm is correct, it is easy to estimate the running
time. Step [Trial division] takes O(n 1/3 ) operations, and if Step [Loop] is
performed, it takes at most
⌈n 1/3 ⌉
k=1
1/6 n
4 √ k +1
= O(n 1/3 )
calls to Algorithm 9.2.11, each call taking O(ln ln n) operations. Thus, in all,
Algorithm 5.1.2 takes in the worst case O(n 1/3 ln ln n) arithmetic operations
with integers the size of n. We now establish the integrity of the Lehman
method.
Theorem 5.1.3. The Lehman method (Algorithm 5.1.2) is correct.
Proof. We may assume that n is not factored in Step [Trial division]. If n
is not prime, then it is the product of 2 primes both bigger than n 1/3 .That
is, n = pq, wherep, q areprimesandn 1/3