Also, I have implemented the above properties and generated the Fibonacci series using c programming language. Skip to main content. Even odd and difference property s of fibonacci numbers International Journal of Development Research.
Kashfia Rahman Oyshei: your comment was deleted, as it revealed a potential solution. Regarding your solution failing at large values of d, look at other comments to the problem for hints. The fibonacci function is in a toolbox. Only functions in vanilla Matlab are recognized by Cody. I think it is a hardware-limited rounding error? When I test eps fibonacci on my system, the answer is 6.
Think very carefully about the number pattern in the Fibonacci sequence, and see if a pattern emerges. Number of 1s in the Binary Representation of a Number. Reindex a vector. Convert a Cell Array into an Array. Pi Digit Probability. Recaman Sequence - II. Van Eck's Sequence's nth member. Happy Free Wednesday! What is the Pisano period of their product AB? Is the Pisano period of A a factor of the Pisano period of B? Look at the complete cycle for any modulus N. It always starts with 0, 1, Here is the complete cycle for 3: mod 3: 0 1 1 2 0 2 2 1 - it has two zeros.
Find a modulus N with some other number of zeros in its cycle. This law is called Benford's Law and appears in many tables of statistics. Other examples are a table of populations of countries, or lengths of rivers. About one-third of countries have a population size which begins with the digit "1" and very few have a population size beginning with "9".
Here is a table of the initial digits as produced by the Fibonacci Calculator : Initial digit frequencies of fib i for i from 1 to Digit: 1 2 3 4 5 6 7 8 9 Frequency: 30 18 13 9 8 6 5 7 4 values Percent: 30 18 13 9 8 6 5 7 4 What are the frequencies for the first Fibonacci numbers or the first 10,?
Are they settling down to fixed values percentages? Use the Fibonacci Calculator to collect the statistics. According to Benford's Law, large numbers of items lead to the following statistics for starting figures for the Fibonacci numbers as well as some natural phenomena Digit: 1 2 3 4 5 6 7 8 9 Percentage: 30 18 13 10 8 7 6 5 5 You do the maths Look at a table of sizes of countries. How many countries areas begin with "1"?
Use a table of population sizes perhaps of cities in your country or of countries in the world. It doesn't matter if the figures are not the latest ones. Does Benford's Law apply to their initial digits? Look at a table of sizes of lakes and find the frequencies of their initial digits.
Using the Fibonacci Calculator make a table of the first digits of powers of 2. Do they follow Benford's Law? What about powers of other numbers? Some newspapers give lists of the prices of various stocks and shares, called "quotations". Select a hundred or so of the quotations or try the first hundred on the page and make a table of the distribution of the leading digits of the prices.
Does it follow Benford's Law? What other sets of statistics can you find which do show Benford's Law? What about the number of the house where the people in your class live? What about the initial digit of their home telephone number? Generate some random numbers of your own and look at the leading digits.
You can buy sided dice bi-pyramids or else you can cut out a decagon a sided polygon with all sides the same length from card and label the sides from 0 to 9. Put a small stick through the centre a used matchstick or a cocktail stick or a small pencil or a ball-point pen so that it can spin easily and falls on one of the sides at random.
See the footnote about dice and spinners on the "The Golden Geometry of the Solid Section or Phi in 3 dimensions" page, for picture and more details.
Are all digits equally likely or does this device show Benford's Law? Use the random number generator on your calculator and make a table of leading-digit frequencies. Such functions will often generate a "random" number between 0 and 1, although some calculators generate a random value from 0 to the maximum size of number on the calculator.
Or you can use the random number generator in the Fibonacci Calculator to both generate the values and count the initial digit frequencies, if you like. Do the frequencies of leading digits of random values conform to Benford's Law? Measure the height of everyone in your class to the nearest centimetre. Plot a graph of their heights.
Are all heights equally likely? Do their initial digits conform to Benford's Law? Suppose you did this for everyone in your school.
Would you expect the same distribution of heights? What about repeatedly tossing five coins all at once and counting the number of heads each time?
What if you did this for 10 coins, or 20? What is the name of this distribution the shape of the frequency graph? When does Benford's Law apply? Random numbers are equally likely to begin with each of the digits 0 to 9. This applies to randomly chosen real numbers or randomly chosen integers. Randomly chosen real numbers If you stick a pin at random on a ruler which is 10cm long and it will fall in each of the 10 sections 0cm-1cm, 1cm-2cm, etc.
Also, if you look at the initial digits of the points chosen so that the initial digit of 0. Randomly chosen integers This also applies if we choose random integers.
Take a pack of playing cards and remove the jokers, tens, jacks and queens, leaving in all aces up to 9 and the kings. Each card will represent a different digit, with a king representing zero. Shuffle the pack and put the first 4 cards in a row to represent a 4 digit integer. Suppose we have King, Five, King, Nine. This will represent "" or the integer whose first digit is 5.
The integer is as likely to begin with 0 a king as 1 an ace or 2 or any other digit up to 9. But if our "integer" began with a king 0 , then we look at the next "digit". These have the same distribution as if we had chosen to put down just 3 cards in a row instead of 4.
The first digits all have the same probability again. If our first two cards had been 0, then we look at the third digit, and the same applies again. So if we ignore the integer 0, any randomly chosen 4 digit integer begins with 1 to 9 with equal probability.
This is not quite true of a row of 5 or more cards if we use an ordinary pack of cards - why? So the question is, why does this all-digits-equally-likely property not apply to the first digits of each of the following: the Fibonacci numbers, the Lucas numbers, populations of countries or towns sizes of lakes prices of shares on the Stock Exchange Whether we measure the size of a country or a lake in square kilometres or square miles or square anything , does not matter - Benford's Law will still apply.
So when is a number random? We often meant that we cannot predict the next value. If we toss a coin, we can never predict if it will be Heads or Tails if we give it a reasonably high flip in the air. Similarly, with throwing a dice - "1" is as likely as "6". Physical methods such as tossing coins or throwing dice or picking numbered balls from a rotating drum as in Lottery games are always unpredictable. Dividing by sqrt 5 will merely adjust the scale - which does not matter.
Similarly, rounding will not affect the overall distribution of the digits in a large sample. Basically, Fibonacci and Lucas numbers are powers of Phi. Many natural statistics are also governed by a power law - the values are related to B i for some base value B.
Such data would seem to include the sizes of lakes and populations of towns as well as non-natural data such as the collection of prices of stocks and shares at any one time. In terms of natural phenomena like lake sizes or heights of mountains the larger values are rare and smaller sizes are more common. So there are very few large lakes, quite a few medium sized lakes and very many little lakes. We can see this with the Fibonacci numbers too: there are 11 Fibonacci numbers in the range , but only one in the next 3 ranges of , , and they get increasingly rarer for large ranges of size The same is true for any other size of range or or whatever.
Clicking the Initial digits button will print the leading digit distribution. Change 1. Does Benford's Law apply here? Washington, The Fibonacci Quarterly vol. Raimi July about his research into Benford's Law. It seems that Simon Newcomb had written about it much earlier, in , in American Journal of Mathematics volume 4, pages The name Benford is, however, the one that is commonly used today for this law.
This is an interesting book but some of the mathematics is at first year university level mathematics or physics degrees , unfortunately, and the rest will need sixth form or college level mathematics beyond age However, it is still good to browse through.
Lucas numbers starting with the number n n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Lucas i starting with n 1 2 3 4 7 11 18 rank i 1 0 2 3 13 23 4 14 19 24 5 10 15 39 20 25 49 6 11 35 Ingenuity in Mathematics Ross Honsberger, Mathematical Association of America , , Essay Six; A Property of a n pages Challenging Mathematical Problems with Elementary Solutions, Vol 1 A M Yaglon and I M Yaglom, Dover paperback edition of the original. It seems we can start off, with the aid of a Calculator, and find Fibonacci numbers ending with all values from 1 to The list starts A Jason Earls comments that there seem to be none that end with , , , , , , Can you prove he is right?
We saw above that when we divide the Fibonacci numbers by a fixed divisor modulus then we always get a cycle of remainders. The length of this cycle is called the Pisano period of the modulus. But is a multiple of 8 so the last 3 digits of any number N bigger than determine the remainder when N itself is divided by 8. So no Fibonacci number of 3 digits or more can end with a 3-digit number which has a remainder of 4 or 6 when divided by 8.
By why do we get Fibonaccis ending with one or two digit numbers whose remainder is 4 or 6 mod 8? The 1 or 2 digit numbers with a remainder of 4 or 6 mod 8 are: 4, 6, 12, 14, 20, 22, 28, 30, 36, 38, 44, 46, 52, 54, 60, 62, 68, 70, 76, 78, 84, 86, 92, 94 However, these can be the endings of larger numbers that are not equivalent to 4 or 6 mod 8, such as a number ending 24 ends in 4 or ends in 6.
So we see that are Fibonacci numbers longer than 2 digits whose one or two digit ending is nevertheless one of the forbidden mod 8 remainders 4 or 6. It happens that we can always find a Fibonacci number ending with any 1 or 2 numbers including those with a remainder or 4 or 6 mod 8: 1 or 2 digit ending 4 6 12 14 20 22 28 30 36 38 44 46 52 54 60 62 68 70 76 78 84 86 92 94 i 9 21 60 93 12 21 36 57 24 45 48 33 18 39 69 Fib i last 3 digits 34 Fib i mod 8 2 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 The Fibonacci Numbers in Pascal's Triangle 0 1 2 3 Each entry in the triangle on the left is the sum of the two numbers above it.
If we re-align the table to look the one on the right then each number is the sum of the one above it and the one to the left of that one where a blank space can be taken as "0". Note that each row starts and ends with "1".
Pascal's Triangle has lots of uses including Calculating probabilities. Why do the Diagonals sum to Fibonacci numbers? It is easy to see that the diagonal sums really are the Fibonacci numbers if we remember that each number in Pascal's triangle is the sum of two numbers in the row above it blank spaces count as zero , so that 6 here is the sum of the two 3's on the row above.
The numbers in any diagonal row are therefore formed from adding numbers in the previous two diagonal rows as we see here where all the blank spaces are zeroes and where we have introduced an extra column of zeros which we will use later: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 The green diagonal sums to 5; the blue diagonal sums to 8; the red diagonal sums to 13 Each red number is the sum of a blue and a green number on the row above.
The sum of all the green numbers is 5 and all the blue numbers add up to 8. Because all the numbers in Pascal's Triangle are made the same way - by adding the two numbers above and to the left on the row above , then we can see that each red number is just the sum of a green number and a blue number and we use up all the blue and green numbers to make all the red ones. The general principle that we have just illustrated is: The sum of the numbers on one diagonal is the sum of the numbers on the previous two diagonals.
Another arrangement of Pascal's Triangle By drawing Pascal's Triangle with all the rows moved over by 1 place, we have a clearer arrangement which shows the Fibonacci numbers as sums of columns: 1 2 3 4 5 6 7 8 9 10 1. It asks how many ways you can pay n pence in the UK using only 1 pence and 2 pence coins. Let's return to Fibonacci's rabbit problem and look at it another way. We shall be returning to it several more times yet in these pages - and each time we will discover something different!
We shall make a family tree of the rabbits but this time we shall be interested only in the females and ignore any males in the population. If you like, in the diagram of the rabbit pairs shown here, assume that the rabbit on the left of each pair is male say and so the other is female.
Now ignore the rabbit on the left in each pair! We will assume that each mating produces exactly one female and perhaps some males too but we only show the females in the diagram on the left.
Also in the diagram on the left we see that each individual rabbit appears several times. For instance, the original brown female was mated with a while male and, since they never die, they both appear once on every line.
Now, in our new family tree diagram, each female rabbit will appear only once. As more rabbits are born, so the Family tree grows adding a new entry for each newly born female. As in an ordinary human family tree, we shall show parents above a line of all their children.
Here is a fictitious human family tree with the names of the relatives shown for a person marked as ME : The diagram shows that: Grandpa Abel and Grandma Mabel are the parents of my Dad; Grandma Freda and Grandpa Fred are the parents of my Mum.
Bob is my Dad's brother my Mum has two sisters, my aunts Hayley and Jane. They have two children, my cousins Sonny Weather and Gale Weather. Gale married Gustof Wind and so is now Gale Wind.
My brother John and his wife Joan have two children, my nephew Dan and my niece Pam. In our rabbit's family tree, rabbits don't marry of course, so we just have the vertical and horizontal lines: The vertical line points from a mother above to the oldest daughter below ; the horizontal line - is drawn between sisters from the oldest on the left down to the youngest on the right; the small letter r represents a young female a little r abbit and the large letter R shows a mature female a big R abbit who can and does mate every month, producing one new daughter each time.
As in Fibonacci's original problem in its variant form that makes it a bit more realistic we assume none die and that each month every mature female rabbit R always produces exactly one female rabbit r we ignore males each month. We can see that our original female becomes a great-grandmother in month 7 when a fourth line is added to the Family tree diagram - a fourth generation! Lots of people submitted solutions to this problem - thank you everyone!
There were too many good solutions to name everybody, but we've used a selection of them below: Most of you decided to start with some examples. Adding three consecutive Fibonacci numbers:. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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