Schnirelmann density
Schnirelmann density is a way to measure how dense a set of numbers is in the whole set of natural numbers (numbers like 1, 2, 3,...). Think of a bowl of M&Ms. If you want to know how many red M&Ms there are, you can count them and divide by the total number of M&Ms in the bowl. That gives you the "density" of red M&Ms.
Now back to numbers. Let's say we have a set of numbers that we're interested in, like all the numbers that can be written as the sum of two prime numbers. We want to know how many numbers in the set are in the whole set of natural numbers. We could count them, but that would take forever!
Instead, we use the Schnirelmann density formula. We take the set of all the numbers in our set and we add them up. Then we divide by the sum of all the numbers in the set of natural numbers up to a certain value (say, 100). This gives us a density value between 0 and 1.
For example, if our set contains the numbers {5, 7, 13, 19, 31}, the sum of the set is 75. The sum of all the numbers from 1 to 100 is 5050, so the Schnirelmann density of our set is 75/5050 = 0.0148.
This means that our set of numbers is not very dense in the whole set of natural numbers. In fact, we can find infinitely many numbers that are not in our set.
So, Schnirelmann density is a way to measure how "thin" or "sparse" a set of numbers is in the whole set of natural numbers. It's like counting M&Ms to see how many are a certain color, but for numbers instead!
Now back to numbers. Let's say we have a set of numbers that we're interested in, like all the numbers that can be written as the sum of two prime numbers. We want to know how many numbers in the set are in the whole set of natural numbers. We could count them, but that would take forever!
Instead, we use the Schnirelmann density formula. We take the set of all the numbers in our set and we add them up. Then we divide by the sum of all the numbers in the set of natural numbers up to a certain value (say, 100). This gives us a density value between 0 and 1.
For example, if our set contains the numbers {5, 7, 13, 19, 31}, the sum of the set is 75. The sum of all the numbers from 1 to 100 is 5050, so the Schnirelmann density of our set is 75/5050 = 0.0148.
This means that our set of numbers is not very dense in the whole set of natural numbers. In fact, we can find infinitely many numbers that are not in our set.
So, Schnirelmann density is a way to measure how "thin" or "sparse" a set of numbers is in the whole set of natural numbers. It's like counting M&Ms to see how many are a certain color, but for numbers instead!