# [seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes

Vladimir Shevelev shevelev at bgu.ac.il
Sat Apr 27 05:21:50 CEST 2013

```The simplest sufficient condition of the periodicity is the following. Let n>3 be such that both numbers p=F(n)+2 and q=F(n+1) are primes. Then Fibonacci-like sequence without multiples of p and q is periodic with period of length n+2. Since then n==1 (mod 3), thus such pairs (p,q) we should seek among only  n==1 (mod 3) . For example, if n=22, then F(22)+2=17713 and F(23)=28657 are both primes. So Fibonacci-like sequence without multiples of 17713 and 28657 is periodic with period of length 24. For what following n>22  F(n)+2 and F(n+1) are both primes?

Best regards,

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Thursday, April 25, 2013 7:21
Subject: [seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

>
> With respect to the question "for a given prime p, whether exist
> one or more  primes greater than p, such that Fibonacci-
> like sequences  without multiples of these (together with
> p) primes is periodic? ", I believe that it should be solved in
> affirmation.On this topic see my sequences A224523 and A219328.
> For example, the Fibonacci-like sequence without multiples of
> primes 659,997,1597,1987 is periodic with period of length 24
> (see comment in A219328). Moreover, up to prime(4000) the
> longest period is of length 198 (see Peter's b-file in A224523).
> But what methods one can use for research of this problem?
>
> Best regards,
>
>
>
> ----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> Date: Monday, April 1, 2013 6:11
> Subject: [seqfan] Periodic Fibonacci-like sequences without
> multiples of several primes
> To: seqfan at list.seqfan.eu
>
> > Dear SeqFans,
> >
> > For a given prime p, consider Fbonacci-like numbers {F_p(n)}
> > without moltiples of p which defined in the following way:
> >  a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2), if a(n-
> > 1)+a(n-2) is not multiple of p, otherwise, a(n)=a(n-1)+a(n-2)
> > divided by maximal possible power of p.
> >  Note that {F_2(n)} has period of length 1, {F_3(n)} has
> > period of length 3, {F_5(n)} has period of length 6. The first
> > case when such a sequence has not a trivial period and,
> > probably, is non-periodic, is p=7 (A224196). I do not know, if
> > this sequence is bounded?
> > Other questions arise when we consider Fibonacci-like
> sequences
> > without multiples of several primes which defined quite
> > analogously: e.g., for {F_(p,q)(n)}, a(0)=0, a(1)=1, for n>=2,
> > a(n)=a(n-1)+a(n-2), if a(n-1)+a(n-2) is multiple of neither p
> no
> > q, otherwise, a(n)=a(n-1)+a(n-2) divided by maximal possible
> > power of p and maximal possible power of q. The first question
> > is the following: for a given prime p, whether exist one or
> more
> > primes greater than p, such that Fibonacci-like sequences
> > without multiples of these (together with p) primes is
> periodic?
> > So in case p=7 we could add primes 11 and 13 and obtain an
> > eventually periodic sequence {F_(7,11,13)(n)} with period of
> > length 12: 0,1,1,2,3,5,8,1,9,10,19,29,48,1,1,2,3,5,...
> > Note that sequence {F_(11,13,19)(n)} has period of length 9,
> > sequence {F_(13,19,23)(n)} has period of length 12, sequence
> > {F_(17,19,23,29)(n)} has period of length 15, sequence
> > {F_(19,23,31,53,59,89)(n)} has period of length 24,
> > while sequence {F_(23,29,73,233)(n)} has period of length 18, etc.
> > By another strange observation, all lengths of periods are
> > multiple of 3 (except for trivial case of {F_2(n)} with period
> > 1). Could anyone to find a periodic Fibonacci-like sequences
> > without multiples of several primes with a length of period
> not
> > multiple of 3?
> >
> > Best regards,
> >
> >  Shevelev Vladimir‎
> >
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> >
>