QUICK! What is the significance of the following sequence of numbers?
1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12 2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4 4 4 12 2 6 6 8
If it helps, here’s the graph of these points (click the graph for a larger picture):
Looking at the graph, you may have been able to figure out that every time a point is at height 2, it corresponds to a prime number. If you got that, then you’ll see that this is a graph of the number of factors each number between 1 and 100 has. Prime numbers, naturally, only have two factors: themselves and the number 1. (The number 1 is itself unique in that it is the only number with only 1 factor: itself).
I made this graph using Excel because, for the past few days, I’ve wondered what it would look like. Don’t ask me why: it’s just one of those things. In any case, as I began to put information into the Excel spreadsheet, there were several patterns that immediately popped out. The way I put in the info was like this.
First, start with a column numbered from 1 to 100. Then include a row from 1 to 100 at the top of the column going from right to left. I then put into the cells the number “1″ in the pattern of “1″ followed by n spaces (where n is the same as the column number). In layman’s terms, this means all the 1′s column is filled with 1s, every other number (the even numbers) is filled in the 2 column, every third number in the 3 column, etc. The result can be seen here (right click it and save it if you wish) or by looking at the following graph (which shows just a portion of the entire 1-100 range, due to size constraints):
First, we notice that listing the numbers in this fashion provides us with a right-triangle:
The slope highlighted in the above is at a 45 degree angle, which shows the upper-limits to the factors of the numbers. There are other lines that show up as well:
In this graphic, the red lines are the prime number slopes.
Naturally, this does look pretty cool. Perhaps there might be a way for someone to figure out the relationships between the primes. I, however, think this is another good illustration of the chaos theory, in this case where well-ordered systems contain hidden chaos. After all, there is no way to predict when the next prime number will occur, although we know some times when it will not–such as on even numbers or numbers that end in 5. Furthermore, there is no way for us to estimate exactly how many factors the next number in the series will have. (In this case, the next number is 101, which I happen to already know is a prime number…but how many factors does 102 have? This cannot be predicted from the graph provided–it has to be calculated.)
Perhaps I will later expand this up to 1,000 (although using Excel for this would be extremely lengthy). In the meantime, I think it’s still just fun to see the strange things one can find in math :-)
