Using pseudoprimes, Carmichael's rule, and Miller's Test how would I go about solving Section 6.2 question 6 6.2 Pseudoprimes2335. Show that if n is an odd composite integer and n is a pseudoprime to the base a, then n is apseudoprime 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 notdividing a (a – 1), then n is a pseudoprime to the base a. Conclude that there are infinitelymany pseudoprimes to any base a. (Hint: To establish that a"-1 = 1 (mod n), show that2p| (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.m8. 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 baseab.10. Suppose that a and n are relatively prime positive integers. Show that if n is a pseudoprimeto 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.%3D12. 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 · 11321%3D

The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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