4. Show that if a²p 1 a² - 1 where a > 1 is an integer and p is an odd prime with pła(a²- 1), n 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
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ISBN:9780470458365
Author:Erwin Kreyszig
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[Number Theory] How do you solve question 4? thanks 

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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