Download Denker Fall 2011 312 Discrete Mathematics

Denker
Fall 2011
312 Discrete Mathematics - Assignment
Date: Friday, 09/23/2011
Problem 1: Find all prime numbers less than 350.
Solution:
2 3 5 7 11 13 17
101 103 107 109
211 223 227 229
307 311 313 317
19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
113 123 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
233 239 241 251 257 263 269 271 277 281 283 293
323 331 337 347 349
Problem 2: Find the prime factorization of 6,177,171.
Solution:
6, 177, 171 = 3 × 7 × 112 × 13 × 187
Problem 3: Let a ≥ 2 be an integeger. Show that if an − 1 is a prime for n ≥ 2, then n
must be a prime.
Solution: Assume n = st is not a prime, s, t ≥ 2. Then
t X
t
n
s
t
(as − 1)k + 1 − 1
a − 1 = ((a − 1 + 1) − 1 =
k
k=1
s
n
so a − 1 divides a − 1, a contradiction. Hence n is prime.
Problem 4: Show that if 2n + 1 is prime for some n ∈ P, then n = 2k for some k ≥ 0.
Solution: Let n = 2k × p2 × p3 × pr be the prime factorization of n. If r ≥ 1 then there
is an odd integer t ≥ 3 with n = 2k t. Hence
t t X
X
t
t
k
2k
l
t−l
t
2k
n
2k
t
(22 +1)l−1 (−1)t−l .
(2 +1) (−1)
+(−1) +1 = (2 +1)
2 +1 = ((2 +1−1) +1 =
l
l
l=1
l=1
Thus 2n + 1 is not prime and r = 0.
Problem 5: Show that 30031 = 2 × 3 × 5 × 7 × 11 × 13 + 1 is not prime, while 2311 =
2 × 3 × 5 × 7 × 11 + 1 is prime.
Solution: A number of the given form cannot be divisible by any of the primes involved
(proof of Thm 1.3.3). Thus the largest prime < 2311 which divides this number must be
smaller than 47. Checking by computer: None of the primes 13, 17,19,23,29,31,37,41,43,47
divides 2311, so the latter is prime.
Running the computer, one finds that 30031 = 59 × 509.