Home primes and foreign primes are produced by a simple recipe that blends prime factorizations with recursion. The home prime of a positive integer n is formed by concatenating the prime factors of n in non-decreasing order. If the resulting integer is prime, then we have found the home prime of n. If not, then we repeat the process as many times as needed to obtain a prime. For instance, 35 = 5·7. After concatenation, we have 57 = 3·19, which is followed by 319 = 11·29, which is followed by 1129, which is prime. Thus, the home prime of 35 is 1129. To obtain the foreign prime of a positive integer n, we form the next integer by concatenating the prime factors of n in nonincreasing order. For example, starting with 35 = 7·5, we next consider 75 = 5·5·3, followed by 553 = 79·7, followed by 797, which is prime. Thus, 797 is the foreign prime of 35. In this talk we give some results about home primes and foreign primes for integers n <100. As one might expect from the arbitrary nature of the concatenation process, there are few easily discernible patterns.
Zoller, Nicholas, "Home Primes and Foreign Primes" (2015). ACMS Conference Proceedings 2015. 23.