The climb to a prime problem asks whether a particular iteration based on prime factorization always reaches
a prime number (Sloane 2017). For a positive
integer 
,
write
 |
where 
are prime numbers and the exponents 
are positive
integers. Define 
by writing each 
followed by 
,
in that order, in base 10 and concatenating
the resulting digits, omitting every exponent
equal to 1. Set 
.
For 
,
2, ..., the values of 
are 1, 2, 3, 22, 5, 23, 7, 23, 32, 25, ... (OEIS A080670).
Starting from 
,
repeatedly apply 
.
If this reaches 1 or a prime number, record that
value, and otherwise record 
. For 
, 2, ..., these values begin 1, 2, 3, 211, 5, 23, 7, 23,
... (OEIS A195264).
The universal prime-reaching claim is false. The composite number
 |
is a fixed point of 
(Sloane 2017). Each of 13, 53, 3853, and 96179 is prime,
and concatenation of 13, 53, 2, 3853, and 96179
gives 
again. Hence 
,
and induction gives 
for every integer 
. This starting value therefore
never reaches a prime number.
This iteration differs from the home prime process, which concatenates repeated
prime factors instead of appending exponents.
For example, 
,
whereas the home prime process starts with 
.
