The 3𝑥+1 Problem as a String Rewriting System
The 3𝑥+1 problem can be viewed, starting with the binary form for any 𝑛∈𝐍, as a string of “runs” of 1s and 0s, using methodology introduced by Błażewicz and Pettorossi in 1983. A simple system of two unary operators rewrites the length of each run, so that each new string represents the next odd int...
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Wiley
2010-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2010/458563 |
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| author | Joseph Sinyor |
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| collection | DOAJ |
| description | The 3𝑥+1 problem can be viewed, starting with the binary form for any 𝑛∈𝐍, as a string of “runs” of 1s and 0s, using methodology introduced by Błażewicz and Pettorossi in 1983. A simple system of two unary operators rewrites the length of each run, so that each new string represents the next odd integer on the 3𝑥+1 path. This approach enables the conjecture to be recast as two assertions. (I) Every odd 𝑛∈𝐍 lies on a distinct 3𝑥+1 trajectory between two Mersenne numbers (2𝑘−1) or their equivalents, in the sense that every integer of the form (4𝑚+1) with 𝑚 being odd is equivalent to 𝑚 because both yield the same successor. (II) If 𝑇𝑟(2𝑘−1)→(2𝑙−1) for any 𝑟,𝑘,𝑙>0, 𝑙<𝑘; that is, the 3𝑥+1 function expressed as a map of 𝑘's is monotonically decreasing, thereby ensuring that the function terminates for every integer. |
| format | Article |
| id | doaj-art-21e6e94071b04e1a968f08f5af834dab |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-21e6e94071b04e1a968f08f5af834dab2025-02-03T01:12:28ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252010-01-01201010.1155/2010/458563458563The 3𝑥+1 Problem as a String Rewriting SystemJoseph Sinyor0Quatronet Corporation, 50 Almond Ave, Thornhill, Ontario, L3T 1L2, CanadaThe 3𝑥+1 problem can be viewed, starting with the binary form for any 𝑛∈𝐍, as a string of “runs” of 1s and 0s, using methodology introduced by Błażewicz and Pettorossi in 1983. A simple system of two unary operators rewrites the length of each run, so that each new string represents the next odd integer on the 3𝑥+1 path. This approach enables the conjecture to be recast as two assertions. (I) Every odd 𝑛∈𝐍 lies on a distinct 3𝑥+1 trajectory between two Mersenne numbers (2𝑘−1) or their equivalents, in the sense that every integer of the form (4𝑚+1) with 𝑚 being odd is equivalent to 𝑚 because both yield the same successor. (II) If 𝑇𝑟(2𝑘−1)→(2𝑙−1) for any 𝑟,𝑘,𝑙>0, 𝑙<𝑘; that is, the 3𝑥+1 function expressed as a map of 𝑘's is monotonically decreasing, thereby ensuring that the function terminates for every integer.http://dx.doi.org/10.1155/2010/458563 |
| spellingShingle | Joseph Sinyor The 3𝑥+1 Problem as a String Rewriting System International Journal of Mathematics and Mathematical Sciences |
| title | The 3𝑥+1 Problem as a String Rewriting System |
| title_full | The 3𝑥+1 Problem as a String Rewriting System |
| title_fullStr | The 3𝑥+1 Problem as a String Rewriting System |
| title_full_unstemmed | The 3𝑥+1 Problem as a String Rewriting System |
| title_short | The 3𝑥+1 Problem as a String Rewriting System |
| title_sort | 3𝑥 1 problem as a string rewriting system |
| url | http://dx.doi.org/10.1155/2010/458563 |
| work_keys_str_mv | AT josephsinyor the3x1problemasastringrewritingsystem AT josephsinyor 3x1problemasastringrewritingsystem |