L(n) are the Lucas numbers: L(n) = F(n−1) + F(n+1) This section was suggested by Peter Bendall, March 2021 Two series in Sloane's Encyclopedia of Integer Sequences are relevant here: A020995 for the index numbers and A067515 for the Fibonacci numbers themselves.On the Sums of Digits of Fibonacci Numbers David Terr, Fibonacci Quarterly, vol.Are there any bases where the Fibonacci numbers with a sum of their base B digits equal to their index numbers form an infinite series? In which bases is it a finite series? However, no proof exists!" A new research question for you to try If you want to try a new investigation, how about converting the Fibonacci numbers to a base other than 10 (binary is base 2 or undecimal is base 11, for example) and seeing what you get for the digit sums in different bases. Thereare no others with N<5000, and it seems likely that Fib(2222) isactually the largest one. Fib(2222) (with 465digits) is the largest known Fibonacci number with this property. This falls further behind N as N gets larger. Thus, unless the digits of Fibonacci numbers have some so-farundiscovered pattern, we would expect the digit sum to be about 0.9 N. Robert Dawson of Saint Mary's University, Nova Scotia,Canada summarises a simple statistical argument (originally in the article referred to below by David Terr) that suggests there may be only a finite number (in fact, just 20 numbers) in this series: "The number of decimal digits in Fib(N) can be shown to be about 0.2 N,and the average value of a decimal digit is (0+1+.+8+9)/10 = 4♵. This makes a nice exercise in computer programming so the computer does the hard work.Ī more difficult question is Does this series (of Fibonacci numbers which have a digit sum equal to their index number) go on for ever? There is also one more whose index number is less than 10 - what is it?Ĭan you find any more in this table of Fibonacci numbersup to Fib(300)?Īs a check, you should be able to find TEN (including those above) up to Fib(200). Here are two more examples of the numbers we seek: Fib(1)=1 and Fib(5)=5. Fib(11)=89 This time the digit sum is 8+9 = 17.īut 89 is not the 17th Fibonacci number, it is the 11th (its index number is 11) so the digit sum of 89 is not equal to its index number.Ĭan you find other Fibonacci numbers with a digit sum equal to its index number? So the index number of Fib(10) is equal to its digit sum. The sum of its digits is 5+5 or 10 and that is also the index number of 55 (10-th in the list of Fibonacci numbers). If we add all the digits of a number we get its digit sum.įind Fibonacci numbers for which the sum of the digits of Fib( n) is equal to its index number n:įor example:- Fib(10)=55 the tenth Fibonacci number is Fib(10) = 55. Digit Sums Michael Semprevivo suggests this investigation for you to try. for the last five digits the cycle length is 150,000.for the last four digits,the cycle length is 15,000 and.For the last three digits, the cycle length is 1,500.
After Fib(300) the last two digits repeat the same sequence again and again. Take a look at the Fibonacci Numbers List or, better, see this list in another browser window, then you can refer to this page and the list together.Ġ, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1 44, 2 33, 3 77, 6 10, 9 87. We can also make the Fibonacci numbers appear in a decimal fraction, introduce you to an easily learned number magic trick that only works with Fibonacci-like series numbers, see how Pythagoras' Theorem and right-angled triangles such as 3-4-5 have connections with the Fibonacci numbers and then give you lots of hints and suggestions for finding more number patterns of your own. We also relate Fibonacci numbers to Pascal's triangle via the original rabbit problem that Fibonacci used to introduce the series we now call by his name. There is an unexpected pattern in the initial digits too. The Mathematical Magic of the Fibonacci Numbers The Mathematical Magic of the Fibonacci Numbers This page looks at some patterns in the Fibonacci numbers themselves, from the digits in the numbers to their factors and multiples and which are prime numbers.
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