Posted on April 29, by The Mathematician Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post. You could come up with a different convention, but why bother?
What about the others? This is true in general. This series on the lower row: It is a subsequence of A above, selecting the numbers at the prime positions.
The smallest Fibonacci number which has the nth prime as a factor gives the series: But as these Fibonacci numbers get large rapidly, it is easier to use the index numbers of such Fibonacci numbers to get the series above A Vajda reference at foot of this pageon page 84 states: An alternative name is the Fibonacci entry point FEP and this applies to any number, not just the primes.
Andrews MacTutor site On another of Ron Knott's Maths pages we look at the Lucas numbers, a series of numbers with the same rule as the Fibonacci numbers but starting with 2 and 1. In many senses Lucas numbers and Fibonacci numbers are twin series. Fibonacci common factors One of the fundamental divisibility properties of Fibonacci numbers concerns factors common to two Fibonacci numbers.
This is a consequence of the formula: If g is the greatest divisor of both F m and F n then it is also a Fibonacci number. Its index number is the greatest divisor common to the two indices m and n!
If we use gcd a,b to mean the greatest common divisor factor of a and b then we have: This section was suggested by an email from Allyn Shell. Neighbouring Fibonacci Numbers have no common factors You might have noticed that no even Fibonacci number is next to another even Fibonacci number, or, no two neighbouring Fibonacci's have a common factor of 2.
The same happens for a common factor of 3, since such Fibonacci's are at every 4-th place Fib 4 is 3.
In fact, there will not be a Fibonacci number as a common factor between two neighbouring Fibonacci's for the same reason. But what about other numbers as factors such as 6 or 7? The answer is that no number bigger than 1 is a factor of two neighbouring Fibonacci numbers. Two numbers that have no common factors are called relatively prime to each other.
There is a proof of this that Tom E Ace wrote to me about -- and it is so simple!
So in any Fibonacci-type series which starts with A and B, if A and B are relatively prime then so are all pairs of consecutive numbers in the series. Now let's look at Fibonacci numbers that have no factors at all apart from 1 and themselves of coursethe prime Fibonacci numbers: This means that if the subscript has factors i.
So what about those Fibonacci numbers with no factors apart from 1 and itself, of course? These are the Fibonacci numbers that are primes. We can now deduce that Any Fibonacci number that is a prime number must also have a subscript that is a prime number again with one little exception - can you find it?
Unfortunately, the converse is not always true: In fact, a search using Maple finds that the list of index numbers, i, for which Fib i is prime begins as follows:Watch breaking news videos, viral videos and original video clips on leslutinsduphoenix.com Subscribe now and save, give a gift subscription or get help with an existing subscription.
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