6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not dividing a(a? many pseudoprimes to any base a. (Hint: To establish that a"-1 = 1 (mod n), show that 2p| (n-1), and demonstrate that a2P = 1 (mod n).) 1), then n is a pseudoprime to the base a. Conclude that there are infinitely doprime to the base 2.
6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not dividing a(a? many pseudoprimes to any base a. (Hint: To establish that a"-1 = 1 (mod n), show that 2p| (n-1), and demonstrate that a2P = 1 (mod n).) 1), then n is a pseudoprime to the base a. Conclude that there are infinitely doprime to the base 2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Using pseudoprimes, Carmichael's rule, and Miller's Test how would I go about solving Section 6.2 question 6
![6.2 Pseudoprimes
233
5. Show that if n is an odd composite integer and n is a pseudoprime to the base a, then n is a
pseudoprime to the base n- a.
6. Show that if n= (a2P - 1)/(a² – 1), where a is an integer, a > 1, and p is an odd prime not
dividing a (a – 1), then n is a pseudoprime to the base a. Conclude that there are infinitely
many pseudoprimes to any base a. (Hint: To establish that a"-1 = 1 (mod n), show that
2p| (n-1), and demonstrate that a2P = 1 (mod n).)
*
|
7. Show that every composite Fermat number F, = 22" + 1 is a pseudoprime to the base 2.
m
8. Show that if p is prime and 2P – 1 is composite, then 2P – 1 is a pseudoprime to the base 2.
-
-
9. Show that if n is a pseudoprime to the bases a and b, then n is also a pseudoprime to the base
ab.
10. Suppose that a and n are relatively prime positive integers. Show that if n is a pseudoprime
to the base a, then n is a pseudoprime to the base ā, where ā is an inverse of a modulo n.
11. Show that if n is a pseudoprime to the base a, but not a pseudoprime to the base b, where
(a, n) = (b, n) = 1, then n is not a pseudoprime to the base ab.
%3D
12. Show that 25 is a strong pseudoprime to the base 7.
13. Show that 1387 is a pseudoprime, but not a strong pseudoprime, to the base 2.
14. Show that 1,373,653 is a strong pseudoprime to both bases 2 and 3.
15/ Show that 25,326,001 is a strong pseudoprime to bases 2, 3, and 5.
16. Show that the following integers are Carmichael numbers.
e) 278,545 = 5· 17 · 29 · 113
21
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfba8c1b-379a-495a-9284-26414a9f3892%2Ff9fa5a80-86b6-4efd-b0ce-1c9f6c319910%2F8u59q4a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6.2 Pseudoprimes
233
5. Show that if n is an odd composite integer and n is a pseudoprime to the base a, then n is a
pseudoprime to the base n- a.
6. Show that if n= (a2P - 1)/(a² – 1), where a is an integer, a > 1, and p is an odd prime not
dividing a (a – 1), then n is a pseudoprime to the base a. Conclude that there are infinitely
many pseudoprimes to any base a. (Hint: To establish that a"-1 = 1 (mod n), show that
2p| (n-1), and demonstrate that a2P = 1 (mod n).)
*
|
7. Show that every composite Fermat number F, = 22" + 1 is a pseudoprime to the base 2.
m
8. Show that if p is prime and 2P – 1 is composite, then 2P – 1 is a pseudoprime to the base 2.
-
-
9. Show that if n is a pseudoprime to the bases a and b, then n is also a pseudoprime to the base
ab.
10. Suppose that a and n are relatively prime positive integers. Show that if n is a pseudoprime
to the base a, then n is a pseudoprime to the base ā, where ā is an inverse of a modulo n.
11. Show that if n is a pseudoprime to the base a, but not a pseudoprime to the base b, where
(a, n) = (b, n) = 1, then n is not a pseudoprime to the base ab.
%3D
12. Show that 25 is a strong pseudoprime to the base 7.
13. Show that 1387 is a pseudoprime, but not a strong pseudoprime, to the base 2.
14. Show that 1,373,653 is a strong pseudoprime to both bases 2 and 3.
15/ Show that 25,326,001 is a strong pseudoprime to bases 2, 3, and 5.
16. Show that the following integers are Carmichael numbers.
e) 278,545 = 5· 17 · 29 · 113
21
%3D
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