The Pythagorean Theorem
The Pythagorean theorem is probably the best known theorem in all of mathematics, or at least among the most popular. It is the famous
a2 + b2= c2
Where a and b are the lengths of the sides of a right triangle and c is the length of the hypotenuse(the side opposite the right angle).
The primitive solution to the Pythagorean problem is:
a2 + b2= c2
Given any two arbitrary integers m and n
a = n2 – m2
b = 2mn
c = m2 + n2
I am not providing a full proof of this solution here. I am simply showing the solution because I use the solution to generate some very interesting patterns.
The Fibonacci Numbers
The Fibonacci sequence is another of the famous topics in mathematics. The Fibonacci sequence is obtained by starting with 0, and 1 and then every other number in the sequence is the result of the previous two numbers. For example: starting with ( 0,1 ) if you add them together you get 0 + 1 = 1 which is then the next number in the sequence leaving us with (0, 1, 1). Then, you can add 1 + 1 = 2 and now our sequence is ( 0, 1, 1, 2 ).
What if we combine the two?
In high school, I was given a task to come up with as many Pythagorean triples as I could using the primitive solutions to the Pythagorean Problem. I always have been a math enthusiast so I thought this project was exciting. So I kept plugging away at the problem trying to make sure I came up with more than anyone else.
When working with the primitive solutions to the Pythagorean problem to make Pythagorean triples I ran into a problem. My problem was it was getting boring just putting random numbers in to get more triples.
So I decided to come up with a more systematic approach to this. I had been reading about Fibonacci, and the famous sequence which bears his name. This gave me the idea that I could use this ready made sequence to substitute into the primitive solution for the Pythagorean problem. This made it less boring than the just using random numbers approach.
I used Fibonacci numbers in the primitive solution and got the following results….
Some of the Pythagorean triples from the chart are:
32 + 42 = 52
52 + 122 = 132
162 + 302 = 342
392 + 802 = 892
Look at the c terms do you see a pattern?……. That’s right the c terms are all Fibonacci numbers.
If you look in the columns under the m and the n you will notice that I have the Fibonacci sequence written in two ways the m column I started the Fibonacci sequence with 0 which still works with the pattern, and in the n column I have the Fibonacci sequence starting with 1. I do this so when I substitute the numbers into the primitive Pythagorean solution They wont just zero out.
There is also another interesting pattern I found with the Fibonacci/Pythagorean triples.
If you look in the two columns on the right I have showed the place value the c terms are in the Fibonacci sequence. Example: the number 1 is the 2nd number in the Fibonacci sequence and the number 2 is the 4th. And this is their place value when you start the Fibonacci sequence with 0.Using the column where I started the Fibonacci sequence with 1 the corresponding c terms are in the odd place values instead.
To explain it another way here I will list out a few of the numbers from the c column
- 1 is the 2nd Fibonacci number when the sequence starts with 0 and the 1st when it starts with 1
- 2 is the 4th Fibonacci number when the sequence starts with 0 and the 3rd when it starts with 1
- 5 is the 6th Fibonacci number when the sequence starts with 0 and the 5th when it starts with 1
- 13 is the 8th Fibonacci number when the sequence starts with 0 and the 7th when it starts with 1
- 34 is the 10th Fibonacci number when the sequence starts with 0 and the 9th when it starts with 1
- 89 is the 12th Fibonacci number when the sequence starts with 0 and the 11th when it starts with 1
- 233 is the 14th Fibonacci number when the sequence starts with 0 and the 15th when it starts with 1
I could probably explain this better,but in general most should be able to look at the chart and see the patterns. There is also more patterns found here when doing this. The one I like is the c column has a lot of prime numbers. a few of them are 2, 5, 13, 89, 233, 1,597, 28,657.
I have carried this out as far as Excel will let me without throwing errors and all of the c terms are Fibonacci Numbers