Home prime
Encyclopedia
In number theory
, the home prime HP(n) of an integer
n greater than 1 is the prime 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 number
s, but the subject is really one in recreational mathematics
.
The outstanding computational problem at present (May, 2011) 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. The pursuit of HP(49) is now a process of factoring a 181-digit composite
factor of HP49(110) after a break was achieved on 8 February 2010 with the calculation of HP49(100) after work lasting the larger part of a decade and an extensive use of computational resources and the successful factorization of a 178-digit composite in HP49(104) into 88- and 90-digit primes on 11 January 2011. 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.
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.
in an article entitled Family Numbers: Constructing Primes By Prime Factor Splicing, which included all of the results HP(n) for n through 100 other than the ones still unresolved. It also included a now-obsolete list of 3-digit unresolved numbers (The 58 listed have been cut precisely in half as of May, 2011). It appears that this article is largely responsible for provoking attempts by others to resolve the case involving 49 and 77. The article uses the terms daughter and parent to describe composites and the primes that they lead to, with numbers leading to the same home prime called siblings (even if one is an iterate of another), and calls the number of iterations required to reach a parent, the persistence of a number
under the map to obtain a home prime, the number of lives. The brief article does little other than state the origins of the subject, define terms, give a couple of examples, mention machinery and methods used at the time, and then provide tables. It appears that Mr. De Geest is responsible for the notation now in use. The OEIS also uses homeliness as the term for the number of numbers, including the prime itself, that have a certain prime as its home prime.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the home prime HP(n) of an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n greater than 1 is the prime obtained by repeatedly factoring
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
the increasing concatenation
Concatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
of prime
Prime
A prime is a natural number that has exactly two distinct natural number divisors: 1 and itself.Prime or PRIME may also refer to:In mathematics:*Prime , the ′ mark, typically used as a suffix...
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 number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s, but the subject is really one in recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
.
The outstanding computational problem at present (May, 2011) 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. The pursuit of HP(49) is now a process of factoring a 181-digit composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
factor of HP49(110) after a break was achieved on 8 February 2010 with the calculation of HP49(100) after work lasting the larger part of a decade and an extensive use of computational resources and the successful factorization of a 178-digit composite in HP49(104) into 88- and 90-digit primes on 11 January 2011. 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
Wiki
A wiki is a website that allows the creation and editing of any number of interlinked web pages via a web browser using a simplified markup language or a WYSIWYG text editor. Wikis are typically powered by wiki software and are often used collaboratively by multiple users. Examples include...
primarily associated with the Great Internet Mersenne Prime Search
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available computer software to search for Mersenne prime numbers. The project was founded by George Woltman, who also wrote the software Prime95 and MPrime for the project...
maintains the complete known data through 1000 in base 10 and also has lists for the bases 2 through 9.
Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of existence
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
of a home prime for any specific number might entail its effective computation. In purely heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
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
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
usually required of mathematical claims.
Early history and additional terminology
While it is unlikely that the idea was not conceived of numerous times in the past, the first reference in print appears to be an article written in 1990 in a small and now-defunct publication called Recreational and Educational Computation. The same person who authored that article, Jeffrey Heleen, revisited the subject in the 1996–7 volume of the Journal of Recreational MathematicsJournal of Recreational Mathematics
The Journal of Recreational Mathematics is an American journal dedicated to recreational mathematics, started in 1968. It is published quarterly by the Baywood Publishing Company....
in an article entitled Family Numbers: Constructing Primes By Prime Factor Splicing, which included all of the results HP(n) for n through 100 other than the ones still unresolved. It also included a now-obsolete list of 3-digit unresolved numbers (The 58 listed have been cut precisely in half as of May, 2011). It appears that this article is largely responsible for provoking attempts by others to resolve the case involving 49 and 77. The article uses the terms daughter and parent to describe composites and the primes that they lead to, with numbers leading to the same home prime called siblings (even if one is an iterate of another), and calls the number of iterations required to reach a parent, the persistence of a number
Persistence of a number
In mathematics, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point, i.e...
under the map to obtain a home prime, the number of lives. The brief article does little other than state the origins of the subject, define terms, give a couple of examples, mention machinery and methods used at the time, and then provide tables. It appears that Mr. De Geest is responsible for the notation now in use. The OEIS also uses homeliness as the term for the number of numbers, including the prime itself, that have a certain prime as its home prime.
See also
- Number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- List of recreational number theory topics
- Prime factorization
- Persistence of a numberPersistence of a numberIn mathematics, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point, i.e...
- Concatenation (mathematics)Concatenation (mathematics)In mathematics, concatenation is the joining of two numbers by their numerals. That is, the concatenation of 123 and 456 is 123456. Concatenation of numbers a and b is denoted a||b. Relevant subjects in recreational mathematics include Smarandache-Wellin numbers, home primes, and Champernowne's...