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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:Specifically, we'll be discussing Pascal's triangle.
Pascal's triangle is an infinite, equilateral triangle composed of numbers.
The numbers that make up Pascal's triangle follow a simple rule: each number is the sum of the two numbers above it. Looking at Pascal's triangle, you'll notice that the top number of the triangle is one. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.
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Pascal’s Triangle is basically a triangle of numbers. “At the tip of the triangle is the number 1, which makes up row zero. Then the second row has two 1’s by adding the 2 numbers above them to the left and right, 1 and 0 (all numbers outside the triangle are zeros).
Pascal's Triangle. One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher)..
To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.