Course notes (chap. 1 Number Theory, chap. 2 ... - McGill University

Algorithm 2.2 ( Kalai randfact(n) )
1: Generate a sequence n = s 0 ≥ s 1 ≥ s 2 ≥ ... ≥ s l =1by picking
s i+1 ∈ R {1, 2,...,s i },untilreachings l =1.
2: Let r be the product of the prime s i ’s, 1 ≤ i ≤ l.
3: IF r ≤ n THEN with probability r/n RETURN (r, {prime s i ’s}).
4: Otherwise, RESTART.
∏
Theorem 2.4 The probability of producing r at step 2 is M n /r, whereM n =
(1 − 1/p).
p≤n
Thus by outputting r with probability r/n in step 3, each possible value
is generated with equal probability M n r
= M n
r n n
.Theoverallprobabilitythat
some small enough r is produced and chosen in step 3 is ∑ M n
1≤r≤n n
= M n .
Theorem 2.5 lim
n→∞
M n log n = e −γ ≈ 0.5614594836