Problem 44 - Discrete Mathematics with Applications

Question

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. \(3,5,\) and 7 are the only three consecutive odd integers that are primes.

Step-by-Step Solution

Answer

We defined three consecutive odd integers as x, x+2, and x+4, where x is an odd number. We analyzed their divisibility by 3, and found that in the cases where x is a multiple of 3 (but not 3 itself) or not, there will always be a composite number among the set of three consecutive odd numbers. Therefore, we conclude that 3, 5, and 7 are the only three consecutive odd integers that are primes.

Step 1: Define the consecutive odd integers

Let's define three consecutive odd integers as x, x+2, and x+4, where x is an odd number.

Step 2: Understanding the prime numbers definition

A prime number is a positive integer greater than 1 that has only two factors, 1 and itself. A composite number is a positive integer greater than 1 that has more than two factors.

Step 3: Consider the divisibility of these consecutive odd integers by 3

Now, if x is an odd number, then it can either be divisible by 3 or not: 1) If x is divisible by 3, then x is a multiple of 3, and so x is not prime (unless x = 3 itself). 2) If x is not divisible by 3, then (x+1) is an even number and not divisible by 3, but (x+2) must be divisible by 3 because there must be a multiple of 3 in every set of three consecutive integers. So, x+2 is not prime.

Step 4: Conclude that there is always a composite number among them

In the case when x is divisible by 3 (and not equal to 3), x is a composite number; in the case when x is not divisible by 3, x+2 is a composite number. So, for all sets of three consecutive odd integers other than (3,5,7), there is at least one composite number, meaning that 3, 5, and 7 are the only three consecutive odd integers that are primes.

Key Concepts

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. They have a remainder of 1 when divided by 2. Common examples of odd integers include 1, 3, 5, 7, 9, and so on. In any sequence of numbers, if you pick every second number starting from one, you will be listing the odd integers.

• Odd integers always end in 1, 3, 5, 7, or 9 when expressed in decimal form.
• Adding or subtracting two odd numbers results in an even number.
• Adding two even numbers also results in an even number, but an odd plus an even equals an odd number.

The uniqueness of the sequence 3, 5, and 7 being odd prime numbers is a rare occurence, highlighting the special pattern among these numbers.

Divisibility

Divisibility refers to the ability of one number to be divided by another without leaving a remainder. For example, 6 is divisible by 2 and 3 since dividing 6 by either number results in a whole number with no remainder.

• If a number is divisible by 1, it means every integer is divisible by 1.
• If a number is divisible by itself, it indicates the number is greater than zero and perfectly divisible.

In the context of consecutive odd integers, divisibility by 3 becomes pivotal. Given three consecutive odd integers, one of them must be divisible by 3. This property helps explain why 3, 5, and 7 are the only consecutive odd integers that are prime. In any other set, at least one number will be composite due to divisibility by 3.

Composite Numbers

Composite numbers are integers greater than 1 that are not prime. This means they have more than two distinct positive divisors, or in simpler terms, they can be divided exactly by numbers other than just 1 and themselves. For example, 4 has divisors 1, 2, and 4, making it composite.

• A composite number always has a factor less than itself beside 1.
• Composite numbers are the opposite of prime numbers, which have only two divisors, one and the number itself.

In the examination of consecutive odd integers, one will be a composite number unless they are the numbers 3, 5, and 7. Larger or smaller sets of consecutive odd integers always end up having at least one integer that is composite due to multiple factors being present.

Consecutive Integers

Consecutive integers are numbers that follow each other in order, without any gaps. They are like sequential steps on a number line. For example, 4, 5, and 6 are consecutive integers, and the pattern continues indefinitely.

• The consecutive odd integers are picked with a difference of 2, for example: 1, 3, 5, 7, etc.
• Given any integer number as a starting point, you can form consecutive numbers by adding 1 or, in the case of the odd series, by adding 2.

In mathematical analysis, understanding properties of these consecutive numbers—especially regarding divisibility and composition—allows us to predict and categorize them. In the case of consecutive odd integers like 3, 5, and 7, these numbers are uniquely aligned to be all prime, a rare set among larger consecutive odd integers which inevitably contain at least one composite number.