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) = 3357, as 10 factors as 2×2×3 yielding HP10(1) = 223, 223 factors as 3×3×5×7 yielding HP10(2) = HP223(1) = 3357, 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 is whether HP(54) and HP(68) 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. The pursuit of HP(54) concerns the factorization of an E3-digit number HP54(62), and the pursuit of HP(68) concerns the factorization of a 112-digit cofactor of HP68(6E). Details of the history of this search, as well as the sequences leading to home primes for all other numbers up to 100, are maintained at Patrick De Geest's worldofnumbers website (HP(n) for all n≤100 except 54 and 68 are all known). A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data up to 1000 in dozenal (base 10) and also has lists for the bases 2 through E.