In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses. Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics.

The outstanding computational problem as of 2016^[update] is whether HP(49) = HP(77) can be calculated in practice. As each iteration is greater than the previous up until a prime is reached, factorizations generally grow more difficult so long as an end is not reached. As of August 2016^[update] the pursuit of HP(49) concerns the factorization of a 251-digit composite factor of HP49(119) after a break was achieved on 3 December 2014 with the calculation of HP49(117).^[1] This followed the factorization of HP49(110) on 8 September 2012^[2] and of HP49(104) on 11 January 2011, and prior calculations extending for the larger part of a decade that made extensive use of computational resources. Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at Patrick De Geest's worldofnumbers website. A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data through 1000 in base 10 and also has lists for the bases 2 through 9.

The primes in HP(n) are

2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, ... (sequence A037274 in the OEIS)

Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of existence of a home prime for any specific number might entail its effective computation. In purely heuristic terms, the existence has probability 1 for all numbers, but such heuristics make assumptions about numbers drawn from a wide variety of processes that, though they are likely correct, fall short of the standard of proof usually required of mathematical claims.