I have a ? that I am kind of stuck on. It says give an infinite list of natural numbers that are not prime. I know half of the numbers are prime and half are not, but I'm not sure how to answer it. Thanks.
There are no natural even numbers that are prime except 2.
To give an infinite list of natural numbers that are not prime, we can start with a simple approach: listing all the natural numbers that are composite (not prime).
A composite number is any natural number greater than 1 that is not prime, i.e., it can be divided evenly by other numbers besides 1 and itself. One way to generate an infinite list of composite numbers is by simply multiplying consecutive natural numbers together.
Let's see how this works with an example:
1. Start with the first composite number, which is 4. We know that 4 = 2 * 2.
2. Move to the next composite number, which is 6. Here, 6 = 2 * 3.
3. Continue this pattern, multiplying consecutive natural numbers to produce more composite numbers:
- 8 = 2 * 2 * 2
- 9 = 3 * 3
- 10 = 2 * 5
- 12 = 2 * 2 * 3
- 14 = 2 * 7
- 15 = 3 * 5
- 16 = 2 * 2 * 2 * 2
- 18 = 2 * 3 * 3
- and so on...
By repeating this process, you can create an infinite list of composite numbers. Remember, composite numbers are any numbers greater than 1 that are not prime, so this approach generates an endless supply of such numbers.
To give an infinite list of natural numbers that are not prime, you can follow these steps:
Step 1: Start with the number 4, which is the first composite number (a number that is not prime) after the number 1.
Step 2: Add 1 to the previous number to get the next number in the list. So the next number would be 5, which is a prime number.
Step 3: Add 1 again to the previous number to get the next number in the list. So the next number would be 6, which is a composite number.
Step 4: Repeat steps 2 and 3 infinitely, alternately adding 1 and skipping the number 1 is prime.
Therefore, an infinite list of natural numbers that are not prime would be: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, ... and so on.