2. Use the method of squaring to compute 51711 mod 1911.
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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Question
[Pseudoprimes] How do you solve #2? The second picture is for context

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 = 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.

Transcribed Image Text:Example 4.5
Let us apply the base 2 test to the integer n = 341. Computing 2341 mod (341)
is greatly simplified by noting that 2¹0 1024 1 mod (341), so
2341 = (2¹0) 342 = 2 mod (341),
and 341 has passed the test. However 341 = 11.31, so it is not a prime but a
pseudo-prime. (In fact, knowing this factorisation in advance, one could 'cheat'
in the base 2 test to avoid large computations: since 11 and 31 are primes,
Theorem 4.3 gives 2¹⁰ = 1 mod (11) and 2³⁰ = 1 mod (31), which easily imply
that 2341 – 2 is divisible by both 11 and 31, and hence by 341.) By checking
that all composite numbers n < 341 fail the base 2 test, one can show that 341
is the smallest pseudo-prime.
=
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