Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat the process. Does this always end in a prime number for any input?
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It's open problem, sequence A037274 from OEIS, socalled "home primes". Hm, the value for n=77 is even unknown. P.S. OnLine Encyclopedia of Integer Sequences definitely should be included in FAQ. 


Is too long for comment: $\ddot\smile$
Although in this specific example there is still a chance to arrive at a prime number, the heuristic is against this conclusion as the length of record increases while prime numbers appear "rarer". 

