Twin Primnes
Problem Statement
A prime number is a whole number greater than 1 whose-number divisors are only 1 and itself. When two prime numbers are 2 units away from one another they are called twin prime numbers. In this activity/POW we were asked to experiment with prime numbers and some of the traits of twin prime numbers. After multiplying twin prime numbers together and adding 1 to the product you should get a number that has these two traits: It should be a perfect square and it should be a multiple of 36.
Although finding values for twin primes and just testing them would be easy our job was to prove why (a*b)+1= a perfect square and a multiple of 36.
Process/Solution
To start the process it is quite simple. I wanted to find twin primes and actually test values of twin primes to see if the the two traits were true. I started testing with numbers around 50 and went up till 100. After solving the equations that I had set up (a*b)+1 I could see that the traits of a perfect square and a multiple of 36 were both visible in the answers. But now I had to prove it. In the beginning I was a little stuck and frustrated because I think that I was trying to solve for one equation. I then refocused and solved to prove each individual trait at a time. To prove the first trait of the perfect square I started by looking at smaller twin prime numbers, 1 and 3, 3 and 5, 5 and 7. I also looked at their values after multiplying and adding 1, their products were 4, 16 and 36 then I immediately caught something. The average of the twin primes was the square root of the final equation, for example:
(1*3)+1=4
1+3=4 4/2=2 and
2*2=4
also...
(5*7)+1=36
5+7=12 12/2=6 and
6*6=36
This pattern worked for all the equations I had set up. Now the question was how can I introduce variables to this to find a proof? So I got started with finding an equation and with a little hint from a friend I got it. Instead of using two variables I used one and that one variable represented the average of the two twin primes. (a*b)+1 = (a-1)(a+1)+1. In this equation the "a" values on the right represent an average so a-1 and a+1 represent the two twin primes. By using FOIL the equation on the right-hand side can be simplified.
(a-1)(a+1)+1
a^2 -1+1
a^2
We can use the numbers 1 and 3 as an example. The average of 1 and 3 is 2 and so 1+1=2 and 3-1=2, in the equation it is written backward. 2-1=1 and 2+1=3. Using this method I can prove that the product of two twin prime numbers +1 can create a perfect square.
The final step of my process was to prove how twin primes +1 can equal a multiple of 36. To start this proof I listed the first 15 multiples 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540. The only multiples of 36 in this list of 15 are 36, 144, 324. In order to prove that (a*b)+1 is a multiple of 36 you need to use the previous equation that we created (a-1)(a+1)+1. To check if a number is a multiple of 36 you divide the number by 36 until you get a whole quotient. So to prove that (a*b)+1 is a multiple of 36 you need to have solved the first equation or at least have the equation and then use the answer in this new equation (a-1)(a+1)+1= the answer /36. After receiving help from a fellow classmate I learned that using an example number that is a square can be helpful. (a-1)(a+1)+1=144/36. In this equation I used 144 as an example. After simplifying you get a^2=4 so a=2. Lets see what we get if we plug 2 into the equation above:
(2-1)(2+1)+1=144/36
1*3+1=144/36
4=4 !!
And that is how you prove why your final value is divisible by 36. Lets try one more:
(a-1)(a+1)+1=324/36
a^2=9
a=3
(3-1)(3+1)+1=324/36
2*4+1=324/36
9=9 !!
Unfortunately 2 and 4 aren't twin primes so this wouldn't work in our case...but you should get the idea.
Self Assessment/Reflection
In my opinion this was an easy problem to understand yourself but it was very hard to describe and articulate it. In this write-up I myself can spot areas of improvement but I described the problem and my solution to the best of my abilities. I did learn a lot from this problem for example the more you know what you are doing and how to do it, the easier it is to tell someone else how to do it or even how to approach it. Initially when people asked me for help I would seem very confusing and they wouldn't understand but after asking my peers and classmates I was able to better understand the problem so I could go back and try and solve the problems for others. In my opinion that is one of the key points of learning and understanding. You need to be able to teach others about what you learned and if you aren't able to teach you haven't learned enough.
A prime number is a whole number greater than 1 whose-number divisors are only 1 and itself. When two prime numbers are 2 units away from one another they are called twin prime numbers. In this activity/POW we were asked to experiment with prime numbers and some of the traits of twin prime numbers. After multiplying twin prime numbers together and adding 1 to the product you should get a number that has these two traits: It should be a perfect square and it should be a multiple of 36.
Although finding values for twin primes and just testing them would be easy our job was to prove why (a*b)+1= a perfect square and a multiple of 36.
Process/Solution
To start the process it is quite simple. I wanted to find twin primes and actually test values of twin primes to see if the the two traits were true. I started testing with numbers around 50 and went up till 100. After solving the equations that I had set up (a*b)+1 I could see that the traits of a perfect square and a multiple of 36 were both visible in the answers. But now I had to prove it. In the beginning I was a little stuck and frustrated because I think that I was trying to solve for one equation. I then refocused and solved to prove each individual trait at a time. To prove the first trait of the perfect square I started by looking at smaller twin prime numbers, 1 and 3, 3 and 5, 5 and 7. I also looked at their values after multiplying and adding 1, their products were 4, 16 and 36 then I immediately caught something. The average of the twin primes was the square root of the final equation, for example:
(1*3)+1=4
1+3=4 4/2=2 and
2*2=4
also...
(5*7)+1=36
5+7=12 12/2=6 and
6*6=36
This pattern worked for all the equations I had set up. Now the question was how can I introduce variables to this to find a proof? So I got started with finding an equation and with a little hint from a friend I got it. Instead of using two variables I used one and that one variable represented the average of the two twin primes. (a*b)+1 = (a-1)(a+1)+1. In this equation the "a" values on the right represent an average so a-1 and a+1 represent the two twin primes. By using FOIL the equation on the right-hand side can be simplified.
(a-1)(a+1)+1
a^2 -1+1
a^2
We can use the numbers 1 and 3 as an example. The average of 1 and 3 is 2 and so 1+1=2 and 3-1=2, in the equation it is written backward. 2-1=1 and 2+1=3. Using this method I can prove that the product of two twin prime numbers +1 can create a perfect square.
The final step of my process was to prove how twin primes +1 can equal a multiple of 36. To start this proof I listed the first 15 multiples 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540. The only multiples of 36 in this list of 15 are 36, 144, 324. In order to prove that (a*b)+1 is a multiple of 36 you need to use the previous equation that we created (a-1)(a+1)+1. To check if a number is a multiple of 36 you divide the number by 36 until you get a whole quotient. So to prove that (a*b)+1 is a multiple of 36 you need to have solved the first equation or at least have the equation and then use the answer in this new equation (a-1)(a+1)+1= the answer /36. After receiving help from a fellow classmate I learned that using an example number that is a square can be helpful. (a-1)(a+1)+1=144/36. In this equation I used 144 as an example. After simplifying you get a^2=4 so a=2. Lets see what we get if we plug 2 into the equation above:
(2-1)(2+1)+1=144/36
1*3+1=144/36
4=4 !!
And that is how you prove why your final value is divisible by 36. Lets try one more:
(a-1)(a+1)+1=324/36
a^2=9
a=3
(3-1)(3+1)+1=324/36
2*4+1=324/36
9=9 !!
Unfortunately 2 and 4 aren't twin primes so this wouldn't work in our case...but you should get the idea.
Self Assessment/Reflection
In my opinion this was an easy problem to understand yourself but it was very hard to describe and articulate it. In this write-up I myself can spot areas of improvement but I described the problem and my solution to the best of my abilities. I did learn a lot from this problem for example the more you know what you are doing and how to do it, the easier it is to tell someone else how to do it or even how to approach it. Initially when people asked me for help I would seem very confusing and they wouldn't understand but after asking my peers and classmates I was able to better understand the problem so I could go back and try and solve the problems for others. In my opinion that is one of the key points of learning and understanding. You need to be able to teach others about what you learned and if you aren't able to teach you haven't learned enough.