Purpose
"A prime number is a whole number greater than 1 whose only whole-number divisors are 1 and itself. When two prime numbers differ by exactly 2, the numbers are called twin primes." Our objective was to find twin primes and use them in the equation (x*y) + 1 = n to find out if the result has two properties: the number is a perfect square and the number is a multiple of 36. We had to experiment with different pairs of twin primes and to find how many twin primes there are in between the numbers 50 and 100.
Process
50-100
I found that there was a total number of three twin primes in between the numbers 50 and 100. First, I found all the possible twin primes and ruled out the ones I knew couldn't truly be prime numbers. I tested the remaining twin primes and narrowed it down to (59,61), (71,73), and (87,89). Using those twin primes, I tested the theory of all twin primes result in a perfect square and multiple of 36 when multiplied together and one being added.
"A prime number is a whole number greater than 1 whose only whole-number divisors are 1 and itself. When two prime numbers differ by exactly 2, the numbers are called twin primes." Our objective was to find twin primes and use them in the equation (x*y) + 1 = n to find out if the result has two properties: the number is a perfect square and the number is a multiple of 36. We had to experiment with different pairs of twin primes and to find how many twin primes there are in between the numbers 50 and 100.
Process
50-100
I found that there was a total number of three twin primes in between the numbers 50 and 100. First, I found all the possible twin primes and ruled out the ones I knew couldn't truly be prime numbers. I tested the remaining twin primes and narrowed it down to (59,61), (71,73), and (87,89). Using those twin primes, I tested the theory of all twin primes result in a perfect square and multiple of 36 when multiplied together and one being added.
My results show that all three prime numbers result in a perfect square, however, only two of them result in a multiple of 36. I thought this was a bit interesting because the theory that twin pairs result in a multiple of 36 isn't true. I tested many other twin primes (some numbers prime and some not) to see if this idea I had was true or not.
As you can see, twin primes do not always result in multiple of 36, but they do seem to always make a perfect square.
Solution
The result is always a perfect square and always result in a multiple of 36. My proof is in the tables I made above to show what is happening with prime numbers. The formula x(x+2)+1 works to prove that prime numbers result in a square and a multiple of 36. For example, (59,61) --> 59(59+2)+1=3600 --> (59*61)+1=3600 --> 3600/36=100 --> √3600=60. The equation proves that twin primes can result in a square and a multiple of 36.
Reflection
I worked very hard on this problem and put a lot of effort into it. I am confident with my process because I showed many examples of twin primes and non twin primes. A habit I think I used well was "Be Confident, Patient and Persistent" because I took my time with this problem to work it out and to see what is really going on. I have many pages of work that show me working out the process, even if it is a bit disorganized. Overall, I am happy with my results and the conclusion I came up with.