MAT 115A
University of California
Fall 2010
Solutions Homework 2
1. Rosen 3.1 #4, pg. 74
Use the sieve of Eratosthenes or Sage to ﬁnd all primes less than 200.
Solution:
The primes less than 200 are 2, 3, 5, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107,
109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181,
191, 193, 197, 199.
2. Rosen 3.1 #5, pg. 74
Find all primes that are the diﬀerence of the fourth powers of two
integers.
Solution:
Let
n
=
x
4

y
4
for
x,y
two integers where
x > y
. Then
n
= (
x

y
)(
x
+
y
)(
x
2
+
y
2
) and is divisible by
x
+
y
which cannot be
1 or
n
. Hence
n
cannot be a prime. So, there are no prime that is the
diﬀerence of the fourth powers of two integers.
3. Rosen 3.1 #11, pg. 74
Let
Q
n
=
p
1
p
2
···
p
n
+1, where
p
1
,p
2
,...,p
n
are the
n
smallest primes.
Determine the smallest prime factor of
Q
n
for
n
≤
6. Do you think that
Q
n
is prime inﬁnitely often? (
Note:
This is an unresolved question.)
Solution:
Q
1
= 3
,Q
2
= 7
,Q
3
= 31
,Q
4
= 211
,Q
5
= 2311
,Q
6
= 30031.
The smallest prime factors are 3,7,31,211,2311, and 59, respectively.
4. Rosen 3.2 #3, pg. 86