In POW# 8 - Twin Primes, we looked at a mathematical phenomenon, twin primes. Twin primes are what it’s called when two prime numbers are separated by one number, like 3 and 5, or 11 and 13. When you multiply twin primes, and then add one, something interesting happens. You always get a perfect square, and, in every case but 3 and 5, you always get a multiple of 36. During this POW, we looked at why this happens, then we proved it with an equation.
For the first part of the problem, we had to look at a bunch of twin primes, their “answers”, so to speak, and then briefly looked at numbers who weren’t twin primes’ behavior in the equation. Here is my data that I used to solve this problem:
For the first part of the problem, we had to look at a bunch of twin primes, their “answers”, so to speak, and then briefly looked at numbers who weren’t twin primes’ behavior in the equation. Here is my data that I used to solve this problem:
A pattern I found is that, going along with the “answer is a perfect square” rule, the perfect square is always the square of the number in the middle of the twin primes. However, when I tested other, non prime, numbers, I found that the “square of the middle number” rule was true in all cases (not just twin primes). Here is my table:
Because of this, I realized that the really unique thing about twin primes was the fact that they were all multiples of 36.
Once I had figured out some important facts about twin primes, I began working on a proof of those facts. That proof came as an equation. It uses variable i, which represents the number in the middle of the twin primes, and also uses a variation on the original j*g+1 equation. Here it is:
Once I had figured out some important facts about twin primes, I began working on a proof of those facts. That proof came as an equation. It uses variable i, which represents the number in the middle of the twin primes, and also uses a variation on the original j*g+1 equation. Here it is:
Unfortunately, the equation works for all numbers, not just twin primes, but it does work. I think one thing I could have done to make the proof more specific was to include 36 in it. I know there is some connecting thread that unifies all of the i values under 36, I just don’t know what it is.
During the problem, I think I used the habit of a mathematician stay organized, because of how I organized my data. I also used generalize, in that I created a generalization to prove my findings.
During the problem, I think I used the habit of a mathematician stay organized, because of how I organized my data. I also used generalize, in that I created a generalization to prove my findings.