Home primes

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The home prime for a given composite number is the prime number reached after iterated reinterpretation as a new number of the previous number's prime factorization stated in ascending order and with multiplicity (a prime number is its own home prime). For example, the home prime for 25 is 773 since $25=5\times 5$, then $55=5\times 11$, then $511=7\times 73$ and 773 is prime. See A037274.

By probabilistic arguments it is known that a prime number is always reached by this process, according to John Conway. However, it is entirely possible that for some rather small numbers the home prime will be far beyond our range of computation. The home prime for 8 is 3331113965338635107, and the home prime for 49 is yet to be found, with calculations stalled just a little past a hundred iterations on an incompletely factored composite number with more than two hundred digits.

Variant with exponents rather than multiplicity

We generally write the factorizations of numbers with exponents rather than multiplicity, e.g., $72=2^{3}\times 3^{2}$ rather than $72=2\times 2\times 2\times 3\times 3$. (The exponent 1 is tacit, thus $360=2^{3}\times 3^{2}\times 5$ rather than $360=2^{3}\times 3^{2}\times 5^{1}$).

Applying this to the home primes, some composite numbers have different home primes. Under this variant, the home prime of 25 is not 773 but 2213, since $25=5^{2}$, then $52=2^{2}\times 13$ and 2213 is prime. (Others have the same home prime, like $4=2\times 2=2^{2}$ and $6=2\times 3$).

Alas, this variant also has its share of numbers with unknown home primes, like 20 and 105. Sean Irvine has calculated the home primes for most numbers up to 1000, and Hans Havermann has already resolved at least five of the question marks. See A195264.

But under this variant, some steps along the way can be significantly smaller than the starting number, such as 3219905755813179726837607 leads to 729 which leads to 36 which ... ending up with 233231 as the home prime. This suggests that the trajectory for some composites may enter a loop (kind of like what happens in one variant of the Collatz problem).