ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Amazing periodicity in prime factors of Fibonacci series, Golden spiral

Updated on September 6, 2013

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!

Magic of the Series
Magic of the Series

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!


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)