Problems & Puzzles: Puzzles

Puzzle 383. Brougnard sequences.

Jacques Tramu sent the following interesting puzzle:

I found in an old paper by G. Brougnard the following sequence of primes (in french "suite de Brougnard"), which may be interesting to your prime puzzlers.

The gb[n]  sequence definition is the following :

1) gb[0] = starter =  any  prime > 2
2) computation of gb[n+1]

let p : = gb[n]
if p  = 2 (mod 3)
         then p  := (p-1)/2  ;
         else p  := p * 2 ;
gb[n+1] := next prime > p  ;

Remark : The sequence 7 17 11 7 .. is periodic.

Definitions:

A gb-sequence is convergent if gb[n] = 7 for some n . The sequence stops at n . Its length is n+1.

A sequence which is not convergent is  divergent .

Examples :

gb[0]=23, then {23, 13, 29, 27, 11, 7}    (length 6)

gb[0]=281, then {281, 149, 79, 163, 331, 673, 1361, 683, 347, 179, 97,  197, 101, 53, 29, 17, 11, 7} (length 18)

pictures: 1, 2

Questions. 

Q1. What is the larger length you can find for a convergent  gb-sequence ?

Q2. What is the larger starter value you can find  for a convergent gb-sequence ?

Q3. What is the larger starter value S you can find such as for any starter <= S all gb-sequences are convergent ?

Q4. Are there sequences gb[0]... gb[n]  such as gb[n] = gb[0] , and gb[0] > 17 ? (periodic sequences)

Q5. Is there an infinite number of convergent sequences ? An infinite number of divergent sequences ?

 

Contributions came from Anton Vrba, B. Boncompagni & J. Wroblewski,

***

Wroblewski wrote:

Regarding Puzzle 383:

Q1.
Farideh suggested me to start with 2501833.
I have used C with gmp mpz_nextprime routine on a 64-bit computer.
The sequence is in:
http://www.math.uni.wroc.pl/~jwr/pp383/f.txt
Note that I have started numbering terms of the sequence from 1 instead
of 0.
Then I was able to find:
http://www.math.uni.wroc.pl/~jwr/pp383/j.txt
which falls into beginning of Farideh's sequence.

My goal is to find a sequence 1,000,000 terms long, so I am trying to
find a sequence which would fall into the begining of the sequence in
j.txt. Unfortunately it will most likely start with a very large term.

Q2-5. I believe that all the sequences are convergent and that there are
no periodic sequences with terms over 17 and moreover I believe noone
will ever prove those claims.

...

Here is my final sequence for Puzzle 383, Question 1.

It comes from putting together 3 sequences:
http://www.math.uni.wroc.pl/~jwr/pp383/m.txt
http://www.math.uni.wroc.pl/~jwr/pp383/j.txt
http://www.math.uni.wroc.pl/~jwr/pp383/f.txt
the last one having starting point suggested by Farideh.

In the files you will find indices together with terms - only terms with
indices divisible by 1000 are reported (also some starting and ending
terms are included).
I have started numbering the terms from 1 instead of 0.

The sequence M starts with
M(1) = 214454067756691347809297997811471503569947533403788989003669094960113749683
080579721707820056537994639493885125453442129276854991981893942069773550759769026
098605936229560221990917712971148299390166534453098783745065431878279697477626132
044803967009588605392355132211787896976995017792743567388330476833866519

The sequence J starts with
J(1) = 22005865332386878933
and we have
J(5) = M(670683) = 352093845318190063331

The sequence F starts with
F(1) = 2501833
and we have
F(5) = J(429980) = 10007401 = M(1100658)

The sequence F ends with
F(390045) = 7 = M(1490698)

Hence we have the sequence M with 1490698 terms.
 

***

B. Boncompagni wrote:

Q1. The sequence starting with 2501833 returns to 7 after 389754 steps
(counting may be wrong by a couple of units due to the process I used to
calculate the sequence) and having reached 311 digits at step 289183.

Q2-Q3. I don't think there are divergent sequences. If we consider the
way we construct a new term, it is clear that the 3-residual may be
considered random. Also, you will get a value which, for large numbers,
is only a tiny bit* more than the double or the half of the previous
term. That means that every second term we have a 50% chance of
returning to (two tiny bits* more than) the same number, and a 25% each
to get it halved or doubled. This is a typical binomial behavior for
which probabilistic analysis says that the chance to get to infinity is
zero: it is logically the same as tossing a coin an infinity of times
and betting on a chance that we will always have more heads than tails.

Q5. That said, the 3-residual might not be random and we might
eventually find a way to construct an infinite sequence: in that case,
we will obviously have an infinity of divergent sequences because every
term in the sequence would start the same infinite sequence. The problem
about the infinity of convergent sequences remains open.

Q4. Because of the binomial behavior and of the fact that we actually
move a tiny bit* forward at every step, the larger the number we start
from, the more unlikely it is to reach a number we have already visited
even with a huge quantity of steps (you will need at least n/log(n)
steps if alternating up and down), so I don't believe any more recurring
sequence exist.

* the average distance between the consecutive primes n-a and n+a is
log(n), so the "tiny bit" should be of the order of magnitude of log(n)/2.

***

Anton wrote:

The question to ask is "do divergent sequences exist at all?"  For example the
sequence starting with the twin prime 1000289 converges in 30 steps but the
sequence starting with its twin, 1000291, finally converges to 17 after 25715
steps and series terms growing as big as 106 digits (at step 16526).

 As there are many similar examples for adjacent primes one could speculate that
every sequence finally converges to 17.  There is no way in determining if a sequence
is divergent other than using  infinite computing power.

***

 


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