# Home prime

The **home prime** of *n*, denoted by HP(*n*) is found by concatenating the prime factors of *n*, repeatedly, until a prime is reached. The notion of a home prime depends on the base (except in the case where *n* itself is prime).

Notations for other bases of 10 are HP_{base}(n) for base *b*.

While it is expected that every *n* in every base has a home prime, experimental evidence indicates that these chains can get quite long.

## Contents

## Base 10

For example, the **home prime** of 25 is 773 because

- 25 = 5 × 5
- 55 = 5 × 11
- 511 = 7 × 73

and finally 773 is prime.

The length of such chains is also of interest, in this case #HP(25) = 3.

## Base 2

HP_{2}(10):

- 10
_{10}= 2 × 5 (10 × 101) - 21
_{10}= 3 × 7 (11 × 111) - 31 prime