Are there infintely many primes generated by the recursion $c_{n+1} = \lceil \frac{3}{2} c_{n} \rceil$?

Inspired by a recent discussion (How to solve a ceiling expression or recurrence equation?) I stumbled on the question:

Are there infinitely many primes in power ceiling series?

If not there must be a largest prime. Then: name it.

A power ceiling series (pcs) is given by a recursion relation (http://mathworld.wolfram.com/PowerCeilings.html)

$$c_{n+1} = \lceil \frac{3}{2} c_{n} \rceil, n=1,2,...\tag{1}$$

with a certain starting value $$c_{1}$$ assumed to be a positive integer.

Empirical study

Instead of attempting an answer along the lines of Euclid (series 1,2,3,...) or Dirichlet (arithmetic progressions) I started to look at some numbers first.

For $$c_{1}=1$$ the series starts like this

$$t_1 = 1,2,3,5,8,12,18,27,41,62,...$$

We identify the primes $$2,3,5, 41$$ in the positions $$2,3,4,9$$ of the series.

At first I thought that there should be no doubt in an affirmtive answer to the question, but let us look at the continuation of the series.

It turns out that the 5th prime comes only in position $$51$$ and is given by $$p_5 = 1'034'394'551$$ which has $$10$$ decimal digits.

Up to $$n=20'000$$ I have found only $$14$$ primes. The largest is in position $$8'331$$ and has $$1'468$$ decimal digits.

The complete list of positions and number of decimal digits in the format (position, number of digits) is:

$$pd = \left( \begin{array}{cc} 2 & 1 \\ 3 & 1 \\ 4 & 1 \\ 9 & 2 \\ 51 & 10 \\ 82 & 15 \\ 137 & 25 \\ 146 & 26 \\ 293 & 52 \\ 497 & 88 \\ 647 & 114 \\ 686 & 121 \\ 6392 & 1126 \\ 8331 & 1468 \\ \end{array} \right) \tag{2}$$

We see that besides the first gap between position $$9$$ and $$51$$, there is another one between $$686$$ and $$6'392$$, and the last one found here between position $$8'331$$ and some position $$\gt 20'000$$.

The first $$12$$ primes in the format $$\text{\{no., prime\}}$$ are

$$\{1,2\} \\ \{2,3\} \\ \{3,5\} \\ \{4,41\} \\ \{5,1034394551\} \\ \{6,297519376459247\} \\ \{7,1440570454126521626927633\} \\ \{8,55380367672992822622688699\} \\ \{9,4253739752907825564688523783014989301745228173158971\} \\ \{10,3559489886026907657946433728467369286058326981306130254381565414153746644560104466249313\} \\ \{11,922734208756764979367278083468852684475364699938688470761168507340945135741698334550793957699227586609814187692457\} \\ \{12,6801985860082123856882760423196279628325420255378234326987284844611806264888577602657321593406159029543493552560633726903\} \\$$

The rareness of primes found so far might point (naively) to a negative answer to the question.

• The sequence elements are often divisible by $2$ or $3$ and in general as numbers get large the probability that they are prime approaches zero, so rarity of primes may be because of that. – kingW3 Oct 5 '18 at 11:06
• The state of knowledge about existence of primes in sparse sets is very poor. In general the answer is "we don't know" unless there is some obvious obstruction implying a "no" answer. The fact the examples you found exist suggest no simple obstruction exists, so you shouldn't expect an answer to be known. – Wojowu Oct 5 '18 at 12:06