Mathematics 400c
Homework (due Mar. 10)
A. Hulpke
37)
(GAP) Find a primitive root modulo 73. Solve the equation 6x ≡ 32 (mod 73).
38)
Determine λ(2268). Find a number a such that ord2268 (a) = λ(2268).
39) Find all solutions to the following equations:
a) 12x ≡ 23 (mod 37)
b) 5x23 ≡ 18 (mod 37)
c) x12 ≡ 11 (mod 37)
40) Suppose that a is a primitive root modulo n and d|ϕ(n).
a) Show that for each x with gcd(x, d) = 1, we have
ϕ(n) ordn a d ·x = d
b) Conclude that for each d|ϕ(n) there are at least ϕ(d) elements of order d.
c) Summing up the numbers of elements obtained in b), show that there are exactly ϕ(d)
elements of order d. (This is independent of n.)
41)
Find a primitive root modulo 195 , and a primitive root modulo 2 · 196 .
42∗ ) Let m be a number with primitive root g. Describe (in terms of g) those a (1 ≤ a < m,
(a, m) = 1) for which the equation x2 ≡ a (mod p) has a solution. How many such a are
there?
Problems marked with a ∗ are bonus problems for extra credit.
Strong Pseudoprimes
The following list (from: C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr., Math. Comp.
35 (1980), no. 151, 1003–1026) gives all composite numbers up to 25 · 109 , that are
simultaneously strong pseudoprime for bases 2, 3 and 5:
25326001, 161304001, 960946321, 1157839381, 3215031751, 3697278427, 5764643587,
6770862367, 14386156093, 15579919981, 18459366157, 19887974881, 21276028621