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.
The primes in HP(n) are (start with n=2)
- 2, 3, 737, 5, 18E194713227E, 7, 2111, 575, 25, E, 3357, 11, 27, 35, 1E59X677360757339047535E15081E, 15, 391, 17, 225, 37, 57, 1E, 10759X5, 511, 737, 18E194713227E, E25, 25, 77101111X2E951079007XE10237E11113E409X254595X1E094X2513681X9342XX0846XX14XE1199357108E031382715704X064E4408XXE05892XE3913X344545915654E7, 27, 7655143E, 3E, 5237, 57, 251345, 31, 217, 575, 8E57733X7E, 35, 237, 37, 1517, E37, 21E, 3E, 33E321, 711, 255, 315, 22177E, 45, 7XX4X597711E1, 5E, 313E8XE5, XE5EE, 225, 4E, 5531, 51, E25, 517X7, ...
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 likely are correct, fall short of the standard of proof usually required of mathematical claims.
- List of recreational number theory topics
- Prime factorization
- Persistence of a number
- Concatenation (mathematics)