# Are there infinitely many prime numbers in the Look-and-say sequence?

Question:

Are there infinitely many prime numbers in the Look-and-say sequence?

In the sequence, I found that $$11$$, $$312211$$ and $$13112221$$ are prime numbers.

I generated the first $$50$$ terms of the sequence; it does not take much time. The prime search is another story. Among the first $$38$$ (above, my computer gave up), as prime numbers, I only found the second and the two you reported.

Congratulations.

• any prime needs certain properties, one of which is the digits that are non zero mod 3, can only exceed the other type of digits non zero mod 3 by a quantity that are 1 or 2 mod 3.
– user645636
Commented Jun 9, 2019 at 20:05
• @RoddyMacPhee. I am very bad in number theory, prime numbers (plus many other areas); so, I feel vry dumb when reading your comment. So, taking into account the manner these numbers are generated, do you have any idea about their primilarity ? Commented Jun 10, 2019 at 2:26
• no, but if we can check how many 1's versus 2's there are we can limit which iterations we ultimately check for primality.
– user645636
Commented Jun 10, 2019 at 9:10
• So is this a hypothesis? And could I name it? Commented Jun 16, 2019 at 3:17
• I should not conclude anything. This was just an observation based on the $38$ first terms. May be, looking at his comments, Roddy MacPhee could help. Commented Jun 16, 2019 at 3:26