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Chapter 4: Problem 44

Show that if \(n\) is prime and \(b\) is a positive integer with \(n \not b,\) then \(n\) passes Miller's test to the base \(b\) .

Short Answer

Expert verified
A prime n passes Miller's test to the base b if \( b^{(n-1)} \bmod n \equiv 1 \), as shown by Fermat's Little Theorem.

Step by step solution

01

- Understand Miller's test

Miller's test involves checking whether a given number, in this case a prime number n, satisfies certain conditions when tested with a base b. The goal is to prove that if n is prime, it will pass Miller's test for any base b where n does not divide b.
02

- Express conditions for n to be prime

Given that n is prime, we need to show that it satisfies the required properties of Miller's test with respect to base b. Recall Miller's test properties include: computing \(b^{(n-1)} \bmod n\), and checking various conditions related to it.
03

- Compute \(b^{(n-1)} \bmod n\) using Fermat's Little Theorem

Fermat's Little Theorem states that for a prime number n and any integer b such that n does not divide b, we have: \[ b^{(n-1)} \bmod n \equiv 1 \]. This simplifies the condition we need to prove.
04

- Verify the condition with the theorem

Using Fermat's Little Theorem, it is clear that if n is a prime number and does not divide the base b, then \[ b^{(n-1)} \bmod n \equiv 1 \]. This indicates that n passes Miller's test for the base b.
05

- Conclude the solution

Since \( b^{(n-1)} \bmod n \equiv 1 \) is satisfied by Fermat's Little Theorem for a prime n, n passes Miller's test to the base b by definition. Therefore, we have proved the required condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number can't be neatly divided by any number other than 1 and the number itself.

Understanding prime numbers is crucial in various fields of mathematics and computer science. They play a key role in cryptography, algorithms, and number theory.

Some important properties of prime numbers include:
  • They have exactly two distinct positive divisors: 1 and the number itself.
  • The number 2 is the smallest and the only even prime number.
  • All other prime numbers are odd.
  • There are infinitely many prime numbers, proven by ancient mathematicians like Euclid.
By understanding prime numbers, we can delve deeper into different mathematical theorems and tests, such as Miller's test, which helps in prime number verification.
Fermat's Little Theorem
Fermat's Little Theorem is a fundamental result in number theory that states: For any prime number n and integer b such that n does not divide b, the following holds true:

\[ b^{(n-1)} \bmod n \equiv 1 \]

In simpler terms, if you take any number b (which is not divisible by n) and raise it to the power of (n-1), then divide it by the prime number n, the remainder will always be 1.

This theorem is incredibly useful in various areas, particularly in cryptography and primality testing:
  • It helps in verifying whether a number is prime.
  • It simplifies calculations involving large exponents.
  • It forms the basis for many advanced algorithms and mathematical proofs.
Using Fermat's Little Theorem in Miller's test helps streamline the process by ensuring that the calculated results always align with the expectations for a prime number.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, called the modulus.

This is similar to the way the hours on a clock reset after reaching 12.

In modular arithmetic, we often use the notation:
\[ a \bmod n \]

To signify the remainder when a integer a is divided by the modulus n.

Some key properties and uses of modular arithmetic include:
  • Simplifying large calculations by reducing numbers within a fixed range.
  • Creating cyclic structures that are helpful in computer science and cryptography.
  • Working with congruences which helps in solving equations and verifying solutions efficiently.
Understanding modular arithmetic is essential for grasping concepts in tests like Miller's test, where we often need to deal with large numbers and their remainders to verify primality.

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Most popular questions from this chapter

Prove or disprove that \(p_{1} p_{2} \cdots p_{n}+1\) is prime for every positive integer \(n,\) where \(p_{1}, p_{2}, \ldots, p_{n}\) are the \(n\) smallest prime numbers.

What is the value of \(\phi\left(p^{k}\right)\) when \(p\) is prime and \(k\) is a positive integer?

Determine whether the integers in each of these sets are pairwise relatively prime. $$\begin{array}{ll}{\text { a) } 21,34,55} & {\text { b) } 14,17,85} \\\ {\text { c) } 25,41,49,64} & {\text { d) } 17,18,19,23}\end{array}$$

Suppose that when a long string of text is encrypted using a shift cipher \(f(p)=(p+k)\) mod 26 , the most common letter in the ciphertext is \(\mathrm{X}\) . What is the most likely value for \(k\) , assuming that the distribution of letters in the text is typical of English text?

Suppose you have intercepted a ciphertext message and when you determine the frequencies of letters in this mes- sage, you find the frequencies are similar to the frequency- of letters in English text. Which type of cipher do you suspect was used?
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