Fibonacci prime number

A Fibonacci Prime (engl. Fibonacci prime ) is a natural number , which at the same time a Fibonacci number and a prime number is. Fibonacci primes are the subject of number theory .

Examples of Fibonacci primes

The sequence of Fibonacci prime numbers begins with the following ten numbers (see sequence A005478 in OEIS ):

${\displaystyle f_{3}=2}$

${\displaystyle f_{4}=3}$

${\displaystyle f_{5}=5}$

${\displaystyle f_{7}=13}$

${\displaystyle f_{11}=89}$

${\displaystyle f_{13}=233}$

${\displaystyle f_{17}=1597}$

${\displaystyle f_{23}=28657}$

${\displaystyle f_{29}=514229}$

${\displaystyle f_{43}=433494437}$

The largest known Fibonacci primes currently known are the following:

${\displaystyle f_{50833},f_{81839},f_{104911}}$

The largest known Fibonacci prime number has 21,925 digits and was discovered by Bouk de Water in April 2001, but was only identified as a prime number by Mathew Steine ​​on October 16, 2015 (as of August 15, 2018). ${\displaystyle f_{104911}}$

There are still much larger numbers that could be Fibonacci prime numbers, but because of their size, one is not yet sure whether they are actually prime numbers or "only" pseudo- prime numbers . In any case, they fulfill many properties of a prime number and it is very likely that they are prime numbers. Such “probable prime numbers” are called PRP numbers . These potential further Fibonacci prime numbers have the following index : ${\displaystyle f_{n}}$ ${\displaystyle n}$

n = 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367

The largest known Fibonacci PRP number has 698,096 digits and was discovered by Henri Lifchitz in March 2018 (as of August 15, 2018). ${\displaystyle f_{3340367}}$

Primality check

There are a number of conditions that can be used in the primality test of Fibonacci numbers and their divisibility properties.

One of these conditions is the following:

For and is a factor of if and only if is a factor of . ${\displaystyle m,n\in \mathbb {N} }$ ${\displaystyle m>2}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle f_{m}}$ ${\displaystyle f_{n}}$

This results in the following condition:

If and is a Fibonacci prime, then is itself a prime. ${\displaystyle n\neq 4}$ ${\displaystyle f_{n}}$ ${\displaystyle n}$

This condition is necessary but not sufficient . Many Fibonacci numbers that have a prime index are not prime. The three smallest example cases are: ${\displaystyle f_{p}}$ ${\displaystyle p}$

${\displaystyle p=19}$ With ${\displaystyle f_{19}=4181=37\cdot 113}$

${\displaystyle p=31}$ With ${\displaystyle f_{31}=1346269=557\cdot 2417}$

${\displaystyle p=37}$ With ${\displaystyle f_{37}=24157817=73\cdot 149\cdot 2221}$

Unsolved problem

One of the big unsolved problems related to the Fibonacci primes is the question:

Are there infinitely many Fibonacci primes?

The Israeli astrophysicist and science author Mario Livio writes:

… So, is there an infinite number of Fibonacci primes…? No one actually knows, and this is probably the greatest unsolved mathematical mystery about Fibonacci numbers.

According to the British mathematician Richard K. Guy, the solution to the problem is very unlikely, he writes:

We are very unlikely to know for sure that the Fibonacci sequence ... contains infinitely many primes.

literature

• Fred Wayne Dodd: Number Theory in the Quadratic Field with Golden Section Unit . 3. Edition. Polygonal Publishing House, Passaic NJ 1983, ISBN 0-936428-08-2 .
• Richard K. Guy: Unsolved Problems in Number Theory . 3. Edition. Springer-Verlag, New York 2004, ISBN 0-387-20860-7 .
• Mario Livio: The Golden Ratio. The Story of Phi, the World's Most Astonishing Number . Broadway Books, New York 2003, ISBN 0-7679-0816-3 .
• Harald Scheid : Number Theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg u. a. 2003, ISBN 3-8274-1365-6 .

Web links

• Eric W. Weisstein : Fibonacci Prime . In: MathWorld (English).

Individual evidence

1. ^ M. Livio: The Golden Ratio. The Story of Phi, the World's Most Astonishing Number . 2003, p. 237 .
2. ^ ^{A b c} R. K. Guy: Unsolved Problems in Number Theory . 2004, p. 17-18 . and also Eric W. Weisstein : Fibonacci Prime . In: MathWorld (English).
3. ↑ The index indicates the position of the respective Fibonacci prime number in the Fibonacci sequence .
4. ↑ f (104911) on Prime Pages
5. ^ Henri Lifchitz, Renaud Lifchitz: PRP Top Records - Search by form F (n). PRP Records, accessed August 14, 2018 . 
6. ^ ^{A b c} F. W. Dodd: Number Theory in the Quadratic Field with Golden Section Unit . 1983, p. 119-120 .