Assignment 3, Math 313
Due: Friday, March 11th, 2016
1 For each of the following integers
(i) determine whether it is prime or composite
(ii) if the integer is composite, factor it using the quadratic sieve
method
• (a) 30941
• (b) 33919
• (c) 68569
• (d) 838861
• (e) 1016801
• (f) 1149847
In each case, choose your
base such that it contains 2 and the first
factor
n
10 odd primes p with p = 1. Choose the value B which determines
the sieving interval large enough that one finds at least 12 values of x
for which x2 − n factors completely over the factor base (this will take
some trial and error).
2 Let us suppose we attempt to factor N = 10553 via the quadratic
sieve. We include all the primes
√ up to 17 in our factor base and test
the values of x within 10 of N . The following values of x lead to
Q(x) which factor over our factor base :
x
Q(x)
93 −24 · 7 · 17
97 −23 · 11 · 13
101
−25 · 11
103
23 · 7
107
27 · 7
110 7 · 13 · 17
111 23 · 13 · 17
Without carrying out any calculations to actually factor N , indicate 6
different quantities which the data suggests might be nontrivial factors
of N .
1
2
3
√ The first four convergents in the continued fraction expansion for
2047 are
pk
45 181 1674 1855
∈
,
,
,
.
qk
1 4
37
41
We may thus write
p2k − 2047qk2 = ±Qk+1
where
Qk+1 = 2 · 11, 32 , 67 and 2 · 32 ,
respectively. Use this information to factor 2047. Show all your work.
What does √
the above say about the period length of the continued
fraction for 2047?
4 Let n = 154421. Factor n using continued fractions. Show all
√ your
work. You will need to use at most the first six convergents to n.