Question

A prime number is defined as an integer greater than 1 whose only positive integer factors are itself and 1. Find two ways to write this definition so that all quantifiers are explicit. (It may be convenient to introduce a variable to stand for the number and perhaps a variable or some variables for its factors.)

Short answer

Primality can be defined as: (1) \( \forall k, (1 < k < n) \Rightarrow (k \nmid n) \) and (2) \( \neg (\exists k, 1 < k < n \) such that \( k \mid n) \).

Step-by-step solution

Introduce the Number Variable

Let the variable \( n \) represent the integer we want to test for primality.

Define the Primality Condition

For the integer \( n \), it is prime if there does not exist an integer \( k \) such that \( k > 1 \) and \( k < n \) and \( k \mid n \). This definition ensures that \( n \) has no divisors other than 1 and itself.

Use Universal Quantifiers

State the condition for \( n \) being prime using explicit quantifiers: \( n \) is prime if \( \forall k, (1 < k < n) \Rightarrow (k mid n) \). This states that for all integers \( k \) between 1 and \( n \), \( k \) should not divide \( n \).

Use Existential Quantifiers

Another way to define \( n \) being prime is to use the negation of the existence of a divisor: \( n \) is prime if \( eg (\exists k, 1 < k < n \) such that \( k \mid n) \). This form states that there does not exist any \( k \) greater than 1 and less than \( n \) that divides \( n \).

Key concepts

Primality Testing

Primality testing is the process of determining whether a given integer is a prime number. Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. To identify if a number is prime, we need to check the factors it can divide evenly without any remainder.
In simplistic terms, if no numbers other than 1 and the number itself divide it evenly, then it's a prime number. For example, the number 7 is only divisible by 1 and 7, so it's prime. In contrast, the number 8 is divisible by 1, 2, 4, and 8, so it's not prime.

• Choose a variable, often termed as 'n', to represent the number you want to test.
• Check all integers between 2 and \(rac{n}{2}\) to see if they divide 'n' evenly.
• If none of these numbers divide 'n', it confirms that 'n' is prime.

Primality testing can be performed using multiple methods, from basic trial division to complex algorithms used in computer programs to quickly verify large numbers.

Integer Factorization

Integer factorization is the process of breaking down a composite number into a product of smaller integers, usually prime numbers. This process is significant in mathematics, especially in number theory, because of how it relates to the properties of numbers.
Factorization involves determining which integers multiply together to yield the original number. If a number is prime, its only factors are 1 and itself, meaning it cannot be factored further.
Here's a simple step-by-step to factor an integer:

• Start with the smallest prime number, 2.
• Divide the integer by 2; if it's divisible, note down 2 as a factor and divide the quotient again by 2, continuing until it's no longer divisible.
• Move to the next prime number (3, 5, 7, etc.), repeating the process until the quotient becomes 1.
• The list of divisors (prime numbers) is the complete factorization of the original integer.

This is a brute force method, which can be computationally costly for very large numbers. Thus, more advanced methods and algorithms are used in practical applications, especially in fields like cryptography.

Quantifiers in Mathematics

Quantifiers are symbols used in mathematics to express the notion of "for all" or "there exists" within a mathematical statement. They help in structuring statements that involve numbers or sets to clarify the conditions that must be met for those statements to be true or false.
The two primary quantifiers are:

• **Universal quantifier (\( \forall \))**: Implies that the statement holds true for all elements in a set or collection. For example, \( \forall x (x > 0)\) indicates that every element, x, in the set is greater than 0.
• **Existential quantifier (\( \exists \))**: Indicates that there is at least one element in the set for which the statement holds true. For example, \( \exists x (x^2 = 4)\) signifies there is at least one x such that x squared equals 4.

Using quantifiers makes mathematical definitions more precise and uniform. As shown in the primality definition example, quantifiers help express complex properties like the absence of factors for a prime number in a clear, concise manner.