Problem 74 - Contemporary Precalculus

Question

Deal with prime numbers. A positive integer greater than 1 is prime if its only positive integer factors are itself and 1. For example, 7 is prime because its only factors are 7 and \(1,\) but 15 is not prime because it has factors other than 15 and 1 (namely, 3 and 5 ). Find the first five terms of the sequence. \(a_{n}\) is the number of prime integers less than \(n\)

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence \(a_n\) are 0, 0, 1, 2, and 3.

Step 1: Identify the prime numbers less than 1:

Considering \(n = 1\), there are no prime numbers less than 1, as a prime number is a positive integer greater than 1. Therefore, \(a_1 = 0\).

Step 2: Identify the prime numbers less than 2:

Considering \(n = 2\), there are no prime numbers less than 2. The first prime number is 2 itself. Therefore, \(a_2 = 0\).

Step 3: Identify the prime numbers less than 3:

Considering \(n = 3\), the prime numbers less than 3 are 2. Therefore, \(a_3 = 1\).

Step 4: Identify the prime numbers less than 4:

Considering \(n = 4\), the prime numbers less than 4 are 2 and 3. Therefore, \(a_4 = 2\).

Step 5: Identify the prime numbers less than 5:

Considering \(n = 5\), the prime numbers less than 5 are 2, 3, and 5. Therefore, \(a_5 = 3\). In conclusion, the first five terms of the sequence \(a_n\) are: \(0, 0, 1, 2,\) and \(3\).

Key Concepts

positive integers

Positive integers are whole numbers that are greater than zero. These numbers do not include fractions, decimals, or negative numbers. They start from 1 and go up to infinity, following a sequence of natural numbers. For example, 1, 2, 3, 4, and 5 are all positive integers.

In the context of prime numbers, a positive integer must be greater than 1 to be considered for primality. This is because 1 only has one distinct factor, which is itself, and does not meet the requirement of having exactly two positive factors.

Understanding positive integers is crucial because they form the basis upon which other mathematical concepts, like integer factors, are built.

integer factors

Integer factors refer to the numbers that can exactly divide a given integer without leaving a remainder. Understanding integer factors is essential for identifying prime numbers.

For any integer, its factors are numbers that, when multiplied together, result in the original integer. For instance, 6 has integer factors 1, 2, 3, and 6 because \(1 \times 6 = 6\) and \(2 \times 3 = 6\).

A number is classified as prime if it only has exactly two distinct positive integer factors: 1 and itself. This means any additional factor, besides 1 and the number itself, disqualifies it from being prime. For example, 15 is not prime because, in addition to 1 and 15, it also has 3 and 5 as factors.

sequences in mathematics

Sequences in mathematics are ordered lists of numbers that follow a specific rule. The sequence mentioned in the exercise is based on finding the count of prime numbers less than a given number \(n\).

Understanding the sequence requires determining how many prime numbers exist below each successive integer \(n\). This sequence builds an intuitive sense of the distribution of prime numbers in the set of positive integers. For example, the first five terms in this specific sequence are 0, 0, 1, 2, and 3, indicating how many prime numbers exist below 1, 2, 3, 4, and 5, respectively.

Such sequences are vital in mathematical studies as they help in pattern recognition, prediction, and analysis of numerical behavior.

prime number identification

Prime number identification is the process of determining whether a given positive integer is prime. This involves checking if the number has exactly two distinct positive integer factors: 1 and the number itself.

To identify a prime number, one can follow these steps:

• Check if the number is greater than 1. Any number less than or equal to 1 cannot be prime.
• Check divisibility by testing factors up to the square root of the number. If the number is only divisible by 1 and itself, it's a prime number.

For instance, to determine if 7 is a prime number, check divisibility by numbers less than its square root (approximately 2.65). Since 7 is not divisible by any integers other than 1 and 7, it is a prime number.

Mastering prime number identification is useful not only in academic exercises but also in fields like cryptography, where prime numbers play a key role.