A Proposal for searching Recurrence Series Primes

Erasmus · 2004-02-14, 12:27

Hi to everyone,

I am interested in searching the general properties of recurrence series, generally
u(1)=p
u(2)=q
u(n+1)=u(n) + u(n-1)
where p and q are either 1 or prime.
Famous examples of them are Fibonacci and Lucas numbers. We already know that there is no proof of finiteness of these numbers being prime.
For now Fibonacci primality has been tested above n=400,000 already. I am both interested in relation degree higher than 3.

The problem can be generalised as a function of f(p1,p2,...,pk,k) where k initial elements are stated and recurrence is developed as follows
u(n+1) = SUM( u(n), u(n-1), ... , u(n+1-k) )

For the moment, my first challenge is to observe the behaviour of Fibonacci-like series with starting elements being changed. I am trying to construct frequency tables for prime numbers among the series, and extract relations between series in terms of primality and factors.

What i have as a problem is the increasing # of digits and my insufficient programming skills. Also i may be short of information about this subject.

Is there anyone that can help or advise some further thoughts for this proposal. Or any known programs that can be used/modified easily to use in this search. My questions may be a little confusing, if somebody is interested, all questions/criticisms are welcome!

Bye

mfgoode · 2004-04-22, 15:55

mfgoode · 2004-04-22, 16:40

Do you mean the tribonacci numbers 1,1,2,4,7,13,24,44,81...(summing of 3 terms) and related series like Lucas series?
It has been shown that the ratio between adjacent numbers as the sequence grows converges on 0.5436890125... the root of the cubic
x^3+x^2 +x-1=0

This series can be generalised by summing four terms (tetranacci numbers), five terms, 6 terms and so on.
In all such sequences the ratio of adjacent terms converges to a limit.
As the number of terms to be summed increases the limiting ratio gets smaller approaching 0.5 as a limit.

In a generalised Fibonacci sequence if the first two no.s are divisible by a prime then all its no.s are divisible by the same prime. This sequence will contain a finite no. of primes.

If the first two nos. are co prime (have no common diviser) the generalised
sequence can contain no primes.
There are infinitely many such sequences but the one with the smallest two starting no.s starts with 34 digit no.s!

Mally.

Erasmus · 2004-05-14, 09:26

Thanks Mally, I appreciate for these infos.

2004-02-14, 12:27	#1
Erasmus Feb 2004 23₁₀ Posts	A Proposal for searching Recurrence Series Primes Hi to everyone, I am interested in searching the general properties of recurrence series, generally u(1)=p u(2)=q u(n+1)=u(n) + u(n-1) where p and q are either 1 or prime. Famous examples of them are Fibonacci and Lucas numbers. We already know that there is no proof of finiteness of these numbers being prime. For now Fibonacci primality has been tested above n=400,000 already. I am both interested in relation degree higher than 3. The problem can be generalised as a function of f(p1,p2,...,pk,k) where k initial elements are stated and recurrence is developed as follows u(n+1) = SUM( u(n), u(n-1), ... , u(n+1-k) ) For the moment, my first challenge is to observe the behaviour of Fibonacci-like series with starting elements being changed. I am trying to construct frequency tables for prime numbers among the series, and extract relations between series in terms of primality and factors. What i have as a problem is the increasing # of digits and my insufficient programming skills. Also i may be short of information about this subject. Is there anyone that can help or advise some further thoughts for this proposal. Or any known programs that can be used/modified easily to use in this search. My questions may be a little confusing, if somebody is interested, all questions/criticisms are welcome! Bye

2004-04-22, 15:55	#2
mfgoode Bronze Medalist Jan 2004 Mumbai,India 804₁₆ Posts	Tribonacci/Tetranacci series and so on.

2004-04-22, 16:40	#3
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²·3³·19 Posts	Tribonacci/tetranacc series and so on Do you mean the tribonacci numbers 1,1,2,4,7,13,24,44,81...(summing of 3 terms) and related series like Lucas series? It has been shown that the ratio between adjacent numbers as the sequence grows converges on 0.5436890125... the root of the cubic x^3+x^2 +x-1=0 This series can be generalised by summing four terms (tetranacci numbers), five terms, 6 terms and so on. In all such sequences the ratio of adjacent terms converges to a limit. As the number of terms to be summed increases the limiting ratio gets smaller approaching 0.5 as a limit. In a generalised Fibonacci sequence if the first two no.s are divisible by a prime then all its no.s are divisible by the same prime. This sequence will contain a finite no. of primes. If the first two nos. are co prime (have no common diviser) the generalised sequence can contain no primes. There are infinitely many such sequences but the one with the smallest two starting no.s starts with 34 digit no.s! Mally.

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2004-05-14, 09:26	#4
Erasmus Feb 2004 27₈ Posts	Thanks Mally, I appreciate for these infos.