Keep factoring and concatenating to get a prime?

Question

Keep factoring and concatenating,starting from $2$ until we get a prime.

$$2=2$$ $$22=2*11$$$$22211=7*19*167$$ $$22211719167=?$$ ...and so on (the prime factors are arranged from smaller to larger and their multiplicities are also written).

How far we can go with these? (because factorization is very hard)

Answer 1

If you have Sage available somewhere, you can run this simple code (n is the initial value, b is the base):

n = 2
b = 10
while True :
    s = n.str(b)
    print n.is_prime(), s
    for p,k in factor(n) :
        s += (p.str(b))*k
    n = Integer(s,b)

The results I managed to get:

True 2
False 22
False 22211
False 22211719167
False 2221171916731111313195493
False 222117191673111131319549333123883568997108723797801
False 222117191673111131319549333123883568997108723797801909138725589671825928623195474482364099822833797133

Needed to say, the sequence either gets constant when you hit a prime, or grows so fast that the probability that there is a prime in $\geq k$th step is moreorless equal to the probability that the $a_k$ is a prime, which is $\frac{\ln a_k}{a_k}$. Good luck.

Answer 2

Using base $8$ arithmetic, it takes two steps:

$2 = 2$

$22 = 2\cdot2\cdot3\quad$ (In base $10$, this is: $\,\,18= 2\cdot2\cdot3$)

$22223\,\,$ is prime. (In base $10$: $\,\,9371$)

On the other hand, as @JMoravitz pointed out, in base $10$ the number gets big. I tried a different site and it too got stuck at $2221171916\ldots133$. It's hard to factor numbers with $104$ digits, but the primality test on the site I used indicates that this number is composite; good luck factoring the next one!

It's hard to answer questions like this because the answer depends very much on the way you choose to represent the numbers (e.g., base $10$) as opposed to any relationship from one number to the next.

Answer 3

If the initial seeds are 3,5,or 7 instead of 2,then we get a prime very quickly,i.e.33311,55511,and 77711 are all primes!. If 2 is the initial seed then that last number 22211719167.....797133 is divisible by 3 eight times(!)and it not divisible by any other primes less than 10^5