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Chapter 4: Problem 20

Determine whether each of these integers is prime, verifying some of Mersenne's claims. $$\begin{array}{ll}{\text { a) } 2^{7}-1} & {\text { b) } 2^{9}-1} \\ {\text { c) } 2^{11}-1} & {\text { d) } 2^{13}-1}\end{array}$$

Short Answer

Expert verified
127 and 8191 are prime, while 511 and 2047 are not.

Step by step solution

01

Calculate 2^7 - 1

First, calculate the value of the expression. 2^7 = 128. Subtract 1 from this value: 2^7 - 1 = 128 - 1 = 127.
02

Determine if 127 is prime

To verify if 127 is a prime number, check for divisibility by prime numbers less than or equal to \(\sqrt{127}\): 2, 3, 5, 7, and 11. 127 is not divisible by any of these. Thus, 127 is prime.
03

Calculate 2^9 - 1

Calculate the value of the expression: 2^9 = 512. Subtract 1 from this value: 2^9 - 1 = 512 - 1 = 511.
04

Determine if 511 is prime

To verify if 511 is a prime number, check for divisibility by prime numbers less than or equal to \(\sqrt{511}\): 511 is divisible by 7 \(511 = 7 \cdot 73\). Thus, 511 is not prime.
05

Calculate 2^11 - 1

Calculate the value of the expression: 2^11 = 2048. Subtract 1 from this value: 2^11 - 1 = 2048 - 1 = 2047.
06

Determine if 2047 is prime

To verify if 2047 is a prime number, check for divisibility by prime numbers less than or equal to \(\sqrt{2047}\): 2047 is divisible by 23 \(2047 = 23 \cdot 89\). Thus, 2047 is not prime.
07

Calculate 2^13 - 1

Calculate the value of the expression: 2^13 = 8192. Subtract 1 from this value: 2^13 - 1 = 8192 - 1 = 8191.
08

Determine if 8191 is prime

To verify if 8191 is a prime number, check for divisibility by prime numbers less than or equal to \(\sqrt{8191}\): 8191 is not divisible by any of these. Thus, 8191 is prime.

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Key Concepts

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

prime numbers
Prime numbers are a special set of numbers. These numbers are only divisible by 1 and themselves. A number is considered prime if it cannot be divided by any other numbers without leaving a remainder.

For example:
  • 2 is prime because it can only be divided by 1 and 2.
  • 5 is also prime because it can only be divided by 1 and 5.


Larger primes are checked in a similar way. Checking a number for primality involves verifying that it is not divisible by any prime numbers less than or equal to its square root. This method is efficient and helps us identify primes quickly.
divisibility rules
Divisibility rules help us determine if one number can be evenly divided by another. These rules are shortcuts to check for factors without performing long divisions.

Here are a few important ones:
  • A number is divisible by 2 if its last digit is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if its last digit is 0 or 5.


To check if larger numbers are primes, these rules can be used to quickly eliminate candidates. For example, for the number 127, we check for divisibility by small primes and find that it isn't divisible by any, confirming it is a prime.
Mersenne numbers
Mersenne numbers are a special type of numbers of the form \(2^n - 1\). These numbers are named after the French mathematician Marin Mersenne.

To be specific, Mersenne primes are a subset of Mersenne numbers that are also prime. For instance, 127 is a Mersenne prime because it is \(2^7 - 1\) and prime.

Here's why they are important:
  • Mersenne numbers quickly grow large and are a key area of study in number theory.
  • Many of the largest known primes are Mersenne primes.


To verify if a Mersenne number is prime, we first compute \(2^n - 1\) and then check its primality using divisibility rules or other primality tests.

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Most popular questions from this chapter

Write an algorithm in pseudocode for generating a sequence of pseudorandom numbers using a linear congruential generator. The middle-square method for generating pseudorandom numbers begins with an \(n\) -digit integer. This number is squared, initial zeros are appended to ensure that the result has 2\(n\) digits, and its middle \(n\) digits are used to form the next number in the sequence. This process is repeated to generate additional terms.

A parking lot has 31 visitor spaces, numbered from 0 to \(30 .\) Visitors are assigned parking spaces using the hashing function \(h(k)=k\) mod \(31,\) where \(k\) is the number formed from the first three digits on a visitor's license plate. a) Which spaces are assigned by the hashing function to cars that have these first three digits on their license plates: \(317,918,007,100,111,310 ?\) b) Describe a procedure visitors should follow to find a free parking space, when the space they are assigned is occupied. Another way to resolve collisions in hashing is to use double hashing. We use an initial hashing function \(h(k)=k \bmod p,\) where \(p\) is prime. We also use a second hashing function \(g(k)=(k+1) \bmod (p-2) .\) When a collision occurs, we use a probing sequence \(h(k, i)=(h(k)+i \cdot g(k)) \bmod p .\)

Let \(p\) be an odd prime and \(r\) a primitive root of \(p .\) Show that if \(a\) and \(b\) are positive integers in \(Z_{p},\) then \(\log _{r}(a b) \equiv\) \(\log _{r} a+\log _{r} b(\bmod p-1)\)

Show that a positive integer is divisible by 11 if and only if the difference of the sum of its decimal digits in even-numbered positions and the sum of its decimal digits in odd-numbered positions is divisible by 11 .

Show that \(a^{m}+1\) is composite if \(a\) and \(m\) are integers greater than 1 and \(m\) is odd. [Hint: Show that \(x+1\) is a factor of the polynomial \(x^{m}+1\) if \(m\) is odd. \(]\)
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