- Education and Science
Amazing periodicity in prime factors of Fibonacci series, Golden spiral
Fibonacci series are numbers such that "every number in the series is sum of its previous two numbers." It goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21 … and so on. The first two digits are called seed value or initial value. The ratio between any two neighboring numbers in this series is said to be the golden ratio". Some claim "this ratio expresses itself everywhere". Some say there exists no golden ratio as such. Is it nowhere or now and here? The following table shows certain beauties of this series now and here!
The series is shown in the middle column. The first column gives serial number to every item of the series. This serial number is important as it demonstrates the periodicity in divisibility of the Fibonacci numbers.
The third number in the series is '2' which is divisible by two. As it is third in the series every next third number in it is divisible by '2'. The fourth number in the series is '3' which is divisible by three. As it is fourth in the series every next fourth number in it is also divisible by '3'.
Seventh number is '13' and therefore 14th, 21st, 28th, and 35th numbers are also divisible by number 13.
Ninth number 34 is divisible by '17' and therefore 18th, 27th and 36th numbers which are- '2584', '196418' and '14930352' respectively are also divisible by '17'.
Tenth number '55' is divisible by '11' and therefore 20th 30th and 40th numbers in the series namely- '6765', '832040' and '102334155' are all divisible by '11'.
Thirteenth number is 233 and, therefore, the 26th and the 39th numbers namely 121393 and 63245986 are also divisible by 233.
Fifteenth number is divisible by '61' therefore 30th number too is divisible by '61'.
Eighteenth number is divisible by '17' and '19' therefore 36th number too is divisible by '17' and '19'.
List of factors given in third column are only the prime factors and the respective Fibonacci number is divisible by many of the multiples of these factors and this law of periodicity holds true for all those multiples as well.
Amazingly this holds true throughout the series. Divisibility of a number in this series depends on the serial number it possesses.
Don't think that your serial number in a list doesn't matter; it determines the characteristics and it is not limited to chemistry only!