Prime Numbers

4.6 Exercises 221
4.14. Suppose that in addition to the hypotheses of Theorem 4.2.10 we
know that all of the prime factors of R1R2 exceed B, wheren−1=F1R1, n+1 = F2R2. Also suppose there is an integer a1 such that a n−1
1 ≡ 1(modn),
gcd(a F1
1 − 1,n) = 1, and there are f,∆ as in (4.12) with gcd(n, 2b) = 1,
∆
n = −1, Un+1 ≡ 0(modn), gcd(UF2,n) = 1. Let F denote the least
common multiple of F1,F2. Show that if the residue n mod F is not a proper
factor of n and BF > √ n,thennis prime.
4.15. Prove Theorem 4.2.9.
4.16. By the methods of Exercise 4.1 show the following: If n>892271479
is prime, let N denote the expected number of choices of random pairs
a, b ∈ {0, 1,...,n − 1}, not both 0, until with f given in (4.12), we have
rf (n) =n +1.ThenN