Twin Primes
Before zI explain my process and explanation I will quickly reflect on what I learned from this POW. To start off we had to find all the primes between 50-100. It was a good refresher to visit primes because I haven't real seen this type of problem in a long time. I also learned about twin primes and how to find a formula to prove the facts given in the problem statement.
PROBLEM STATEMENT/PROCESS
When you multiply twin primes and add 1, you get:
- The number is a perfect square number
- The number is a factor of 36
What this POW is asking us is to prove that these theories are correct and work. So to start off with proving that the number becomes a perfect square we must create a formula. I had trouble finding one so a classmate and I found the formula: (a-1) (a+1) +1 =a* (in this case, *=sqaured) So the first thing I did was to try the formula and plug in a number. I used 3 and this is the proof that it works:
(a-1) (a+1) +1 = a*
(3-1) (3+1) +1 =9*
(2) (4) +1 = 9*
8 + 1= 9
9=9
(a-1) (a+1) +1 = a*
(7-1) (7+1) +1 =49*1
(6) (8) +1 = 49*
48 + 1= 49
49=49
but to prove that using this equation is correct if we use a* it'll prove that it's a correct function./36
a* + a-a-1+1=a*
a*+0+0=0*
a*+0=a*
a*=a*
Now that we have proved that the number alway become a perfect square we must show that it must be divisible by 36. But every prime is divisible by 6, therefore should be divisible by 36. I'm not as confident in this proof then I was in the one before but this is what I found.
When using this equation: multiply primes and add 1, we showed that it comes out to a square number which is then easier to divide by 6 or 36. To show that it works we csn use the prime numbers 5 and 7.
5x7+1=36
36/36= 14
now we can use the numbers 11 and 13
11x13+1= 144
144/36= 4,.
If we kept using prime numbers we would keep getting factors of 36 just like we did with 1 and 4.
PROBLEM STATEMENT/PROCESS
When you multiply twin primes and add 1, you get:
- The number is a perfect square number
- The number is a factor of 36
What this POW is asking us is to prove that these theories are correct and work. So to start off with proving that the number becomes a perfect square we must create a formula. I had trouble finding one so a classmate and I found the formula: (a-1) (a+1) +1 =a* (in this case, *=sqaured) So the first thing I did was to try the formula and plug in a number. I used 3 and this is the proof that it works:
(a-1) (a+1) +1 = a*
(3-1) (3+1) +1 =9*
(2) (4) +1 = 9*
8 + 1= 9
9=9
(a-1) (a+1) +1 = a*
(7-1) (7+1) +1 =49*1
(6) (8) +1 = 49*
48 + 1= 49
49=49
but to prove that using this equation is correct if we use a* it'll prove that it's a correct function./36
a* + a-a-1+1=a*
a*+0+0=0*
a*+0=a*
a*=a*
Now that we have proved that the number alway become a perfect square we must show that it must be divisible by 36. But every prime is divisible by 6, therefore should be divisible by 36. I'm not as confident in this proof then I was in the one before but this is what I found.
When using this equation: multiply primes and add 1, we showed that it comes out to a square number which is then easier to divide by 6 or 36. To show that it works we csn use the prime numbers 5 and 7.
5x7+1=36
36/36= 14
now we can use the numbers 11 and 13
11x13+1= 144
144/36= 4,.
If we kept using prime numbers we would keep getting factors of 36 just like we did with 1 and 4.