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1. Show that 25 is a strong pseudoprime base 7, i.e., passes Miller's test base 7. 2. Use the method of squaring to compute 51711 mod 1911. 3. (i) For a prime p suppose that n = 2º - 1 is not a prime. Show that n is a pseudoprime base 2. (ii) Show every composite Fermat number Fm 4. Show that if = n 22m + 1 is a pseudoprime base 2. a²p 1 a² 1 where a > 1 is an integer and p is an odd prime with pła(a²-1), then n is a pseudoprime base a. Hint: First show that 2p | (n-1). Then consider an-1 – 1. - Note that this shows that there are infinitely many pseudoprimes base a for any a.

1. Show that 25 is a strong pseudoprime base 7, i.e., passes Miller's test base 7. 2. Use the method of squaring to compute 51711 mod 1911. 3. (i) For a prime p suppose that n = 2º - 1 is not a prime. Show that n is a pseudoprime base 2. (ii) Show every composite Fermat number Fm 4. Show that if = n 22m + 1 is a pseudoprime base 2. a²p 1 a² 1 where a > 1 is an integer and p is an odd prime with pła(a²-1), then n is a pseudoprime base a. Hint: First show that 2p | (n-1). Then consider an-1 – 1. - Note that this shows that there are infinitely many pseudoprimes base a for any a.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Show that 25 is a strong pseudoprime base 7, i.e., passes Miller's test base 7. 2. Use the method of squaring to compute 51711 mod 1911. 3. (i) For a prime p suppose that n = 2² - 1 is not a prime. Show that n is a pseudoprime base 2. (ii) Show every composite Fermat number Fm 4. Show that if = n 22m + 1 is a pseudoprime base 2. a²p 1 a² - 1 where a > 1 is an integer and p is an odd prime with pła(a²-1), then n is a pseudoprime base a. Hint: First show that 2p | (n − 1). Then consider an−¹ – 1. Note that this shows that there are infinitely many pseudoprimes base a for any a.
Transcribed Image Text:1. Show that 25 is a strong pseudoprime base 7, i.e., passes Miller's test base 7. 2. Use the method of squaring to compute 51711 mod 1911. 3. (i) For a prime p suppose that n = 2² - 1 is not a prime. Show that n is a pseudoprime base 2. (ii) Show every composite Fermat number Fm 4. Show that if = n 22m + 1 is a pseudoprime base 2. a²p 1 a² - 1 where a > 1 is an integer and p is an odd prime with pła(a²-1), then n is a pseudoprime base a. Hint: First show that 2p | (n − 1). Then consider an−¹ – 1. Note that this shows that there are infinitely many pseudoprimes base a for any a.
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