Question

Use the method of direct proof to prove the following statements. If \(n \in \mathbb{N}\) and \(n \geq 2,\) then the numbers \(n !+2, n !+3, n !+4, n !+5, \ldots, n !+n\) are all composite. (Thus for any \(n \geq 2,\) one can find \(n-1\) consecutive composite numbers. This means there are arbitrarily large "gaps" between prime numbers.)

Short answer

The statement is proven by showing that each term in the sequence \(n!+2, n!+3, ..., n!+n\) is composite, meaning each of these terms has more than two distinct positive divisors. Since \(n!\) incorporates each integer from 2 to \(n\) in its multiplication, adding these integers subsequently to \(n!\) will result in numbers that are divisible by this added integer - proving them to be composite.

Step-by-step solution

Understand the Statement

The problem suggests that for every integer \(n\) that is greater than or equal to 2 if you add any number from 2 to \(n\) to \(n!\), the result will be a composite number. A composite number is an integer that has more than two different positive divisors, which is contrary to a prime number that has exactly two distinct positive divisors.

Break Down the Problem

The statement to be proven can be broken down into separate parts. First, the term \(n!+2\) can be understood as \(n!\) adding 2. Because \(n!\) is the product of all positive integers less than or equal to \(n\), it means that 2 divides \(n!\). Then, adding 2 to \(n!\), the resulting number \(n!+2\) is divisible by 2 and therefore not a prime number, confirming it to be a composite number. This logic can be applied to all the succeeding terms from \(n!+3\) to \(n!+n\).

Direct Proof

This proof appropriately uses principles of arithmetic and the characteristics of prime and composite numbers. For \(n!+2\), it is clear that this number is divisible by 2 since \(n!\) is divisible by 2. Proceeding with any integer from 3 to \(n\), it can be seen that adding this integer to \(n!\), always results in a number divisible by the chosen integer, as \(n!\) previously multiplied this integer. Therefore, all numbers in the sequence are composite.

Key concepts

Understanding Composite Numbers

When delving into the intricacies of mathematics, composite numbers prove themselves to be a fundamental concept. A composite number is defined as any integer greater than 1 that is not a prime number. This means that a composite number has more than two distinct positive divisors.

For example, let's consider the number 6. It can be divided by 1, 2, 3, and 6. Since it has four divisors, it is classified as composite. On the other hand, the number 3 is only divisible by 1 and itself, making it a prime number rather than composite.

In a way, you can think of composite numbers as a kind of 'team' of factors working together, as opposed to prime numbers which stand alone with their single distinct pair of factors. This description helps in visualizing why numbers like 2, 3, 5, 7, and so on are primes (they only have two divisors–1 and the number itself), while others like 4 (with divisors 1, 2, and 4) or 8 (with divisors 1, 2, 4, and 8) are composites.

The Factorial Function and Its Significance

At its core, the factorial is a function that takes a nonnegative integer and multiplies it by all positive integers less than itself. Denoted as 'n!' for a given integer 'n', this process creates an increasingly larger product as 'n' becomes bigger.

For instance, the factorial of 3, written as '3!', is calculated as 3 * 2 * 1, which equals 6. Similarly, '5!' would be the product of 5 * 4 * 3 * 2 * 1, resulting in 120. It's evident that factorials grow extremely rapidly—a property that becomes vital in various mathematical domains such as permutations, combinations, and series expansions.

The factorial function is intrinsically linked with the concepts of permutations and combinations, which are the different ways items can be arranged or selected, respectively. In permutations, 'n!' represents the total number of ways 'n' distinct items can be arranged, while in combinations, it is used to calculate the number of ways to select a subset of items from a larger set.

Prime Numbers Unveiled

Prime numbers, the building blocks of the natural numbers, are often described as the 'atoms' of the mathematical world. They are defined as natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

Examples of prime numbers include 2, 3, 5, 7, and 11. Each one is unique in that it cannot be formed by multiplying two smaller natural numbers together. In contrast to composite numbers, which can be decomposed into smaller factors, prime numbers stand indivisible.

Prime numbers hold a special place in number theory because of their fundamental properties and their usage in algorithms and cryptographic systems. They are not just theoretical constructs; their indivisibility makes them practical tools. For example, the security of many encryption algorithms relies on the difficulty of factoring the product of two large prime numbers, showcasing one of the real-world applications of this abstract mathematical concept.