## Problem statement:

A prime number is a whole that is greater than 1 and is only divisible 1 and itself. A twin prime is a set of two numbers that exactly differs by 2. Examples of a twin primes are 5 and 7, 17 and 19, and 41 and 43. As you move forward with twin primes you find this phenomenon that is true for any pair of twin primes, with the exception of 3 and 5. If you multiply the twin primes and add one to the product you a get number that has these two properties.

Keeping this phenomenon in mind experiment with different twin primes and try to find why this phenomenon works the way it does; prove the two facts about multiplying twin primes and adding one.

- The final product is a perfect square.
- The number is a multiple of 36.

Keeping this phenomenon in mind experiment with different twin primes and try to find why this phenomenon works the way it does; prove the two facts about multiplying twin primes and adding one.

## process/Solution:

As I tackled this POW, I right away started listing the twin primes and plugging them into this "phenomenon". As part of my breakdown process, I started to list out the twin primes between 50 and 100. As soon as I started checking the products of the twin primes for the properties of the "Twin Prime Phenomenon" I was thrown off the course I was on. A very good chunk of the numbers did not fit into the criteria; I went back to my initial work and reviewed my math. After this little push back I decided to set goals for myself, even if I didn't find an "absolute" answer I could at least convince myself. My first goal was to prove why the numbers were perfect squares, after a bit of collaboration with peers I found that using the average of the two Twin Primes you would receive a perfect square. My next goal was to find or at least test an equation that would prove that the result is always a perfect square. The equation I found that proved the perfect square concept was

**(X[X+2])+1.**

Thanks to many examples that were tested and all the numbers that were plugged in the equation above proved that the product of the twin primes are perfect squares.## Self grade/reflection:

Working through this POW I struggled at times, out of all the POWs from this school year I found this one the most difficult. This POW is based off of experimentation with numbers. Experimenting through conjectures played a very big role in my process to understanding what this problem was asking for. I have found that plugging in numbers works best for me, it really illustrates a vivid picture of what direction to take in math; I like to have a solid foundation/starting point and base the following steps with trial and error. This POW in specific really opened my eyes to the benefit of having someone to you through a thinking process; collaboration was a very significant factor in my equation conjecture phase. When I was collaborating with others I saw that just talking, bouncing ideas, and findings really help you see things you missed before. Having a fresh set of eyes can really help you find simple mistakes and significant findings you may overlooked. In all this POW may have caused some headaches, but the new critical thinking skills I gained in the process were so worth it.