Math 4150 Number Theory
Homework 5 Solutions
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Section 6.2 #12,13,18ab
Section 8.4 #6,8,13,14
Section 8.6 #1,2,6ab
6.2.12 Grading scheme:
0.5 pts: Wrote something but did not show that Miller’s test was satisfied and did not
state that 25 is a strong pseudoprime base 7.
1 pt.: Showed some work “related to” Miller’s test and stated that 25 is a strong
pseudoprime base 7 but did not show that 7^(2*3) is congruent to -1 (mod 25)
1.5 pt: Showed that 7^(2*3) is congruent to -1 (mod 25) and stated that 25 is a strong
pseudoprime base 7 but work had some errors or lacked detail.
2 pt: Correctly used Miller’s test:
A composite number n = d · 2s + 1 with d odd is called a strong pseudoprime to a
relatively prime base a iff one of the following conditions hold:
In this problem:
a=7
n=25=3*2^3-1
d=3
s=3
The first condition does not hold: 7^3 is not congruent to 1 mod 25.
So we have to see if the next condition holds. Taking r=1, 7^(2*3) is congruent to -1
(mod 25) so that condition holds, and 25 is a strong pseudoprime to the base 7 by Miller’s
test.