Originally, wild numbers are the numbers supposed to belong to a fictional sequence of numbers imagined to exist in the mathematical world of the mathematical fiction The Wild Numbers authored by Philibert Schogt, a Dutch philosopher and mathematician.[1] Even though Schogt has given a definition of the wild number sequence in his novel, it is couched in a deliberately imprecise language that the definition turns out to be no definition at all. However, the author claims that the first few members of the sequence are 11, 67, 2, 4769, 67. Later, inspired by this wild and erratic behaviour of the fictional wild numbers, American mathematician J. C. Lagarias used the terminology to describe a precisely defined sequence of integers which shows somewhat similar wild and erratic behaviour. Lagaria's wild numbers are connected with the Collatz conjecture and the concept of the 3x + 1 semigroup.[2] [3] The original fictional sequence of wild numbers has found a place in the On-Line Encyclopedia of Integer Sequences.[4]
In the novel The Wild Numbers, The Wild Number Problem is formulated as follows:
But it has not been specified what those "deceptively simple operations" are. Consequently, there is no way of knowing how those numbers 11, 67, etc. were obtained and no way of finding what the next wild number would be.
The novel The Wild Numbers has constructed a fictitious history for The Wild Number Problem. The important milestones in this history can be summarised as follows.
| Date | Event | |
|---|---|---|
| 1823 | Anatole Millechamps de Beauregard poses the Wild Number Problem in its original form. | |
| 1830s | The problem is generalised: How many wild numbers are there? Are there infinitely many wild numbers? It was conjectured that all numbers are wild. | |
| 1907 | Heinrich Riedel disproves the conjecture by showing that 3 is not a wild number. Later he also proves that there are infinitely many non-wild numbers. | |
| Early 1960s | Dimitri Arkanov sparks renewed interest in the almost forgotten problem by discovering a fundamental relationship between wild numbers and prime numbers. | |
| The present | Isaac Swift finds a solution. |
In mathematics, the multiplicative semigroup, denoted by W0, generated by the set
| \left\{ | 3n+2 |
| 2n+1 |
:n\geq0\right\}
\left\{
| 1 | |
| 2 |
\right\}\cup\left\{
| 3n+2 | |
| 2n+1 |
:n\geq0\right\}
The On-Line Encyclopedia of Integer Sequences (OEIS) has an entry with the identifying number relating to the wild numbers. According to OEIS, "apparently these are completely fictional and there is no mathematical explanation". However, the OEIS has some entries relating to pseudo-wild numbers carrying well-defined mathematical explanations.[4]
Even though the sequence of wild numbers is entirely fictional, several mathematicians have tried to find rules that would generate the sequence of the fictional wild numbers. All these attempts have resulted in failures. However, in the process, certain new sequences of integers were created having similar wild and erratic behavior. These well-defined sequences are referred to as sequences of pseudo-wild numbers. A good example of this is the one discovered by the Dutch mathematician Floor van Lamoen. This sequence is defined as follows:[7] [8]
For a rational number p/q let
f(p/q)=
| pq | |
| sumofdigitsofpandq |
For a positive integer n, the n-th pseudo-wild number is the number obtained by iterating f, starting at n/1, until an integer is reached, and if no integer is reached the pseudo-wild number is 0.
For example, taking n=2, we have
| 2 | |
| 1 |
,
| 2 | |
| 3 |
,
| 6 | |
| 5 |
,
| 30 | |
| 11 |
,66
and so the second pseudo-wild number is 66. The first few pseudo-wild numbers are
66, 66, 462, 180, 66, 31395, 714, 72, 9, 5.