In 2013-2014, author (click here) wrote natural numbers in increasing and decreasing orders of 1 to 9 using only basic operations. See below some examples:
This work has been considered as “Improbable Research”. For details see the links:
Also refer Nebus 1, Nebus 2 for more comments. We observe that the number 10958 is the only number among 0 to 11111 that is not possible to write in terms of 1 to 9 in the increasing case. We need to use extra operations such as square-root and/or factorial to write it. See below:
Numberphile produced two videos on YouTube using concatenation as an extra operation to bring representation of number 10958 in increasing order. See below both the links:
Total views: 11.644.599 till 22.10.21
Also this number has become a question of discussion at various web-sites. See the links below:
- Extension to 11112-20000 – https://doi.org/10.5281/zenodo.2543626;
- Extension to 20001-25000 – Taneja-2 ;
- Extension to 25001-30000 – Taneja-3.
- Revised: 20001-3000: Crazy Representations of Natural Numbers From 20001 to 30000, Zenodo, February 3, 2019, pp. 1-276, http://doi.org/10.5281/zenodo.2556036.
- By use of different operations, Bill Ames (AMES1, AMES2), recently, extended the above representations up to 1,111,111 in ascending orders of 1 to 9. The first link AMES1 is YouTube vídeo, and the second link AMES2 is txt file.
- Also extension to small bases refer:T. WYLIE, Crazy Sequential Representations of Numbers for Small Bases.
- Anne E. Bras and V.H.J. van der Velden extended the results from – 2147483647 up to 214748364 using only the basic operations. As the numbers increases the gaps also increases. For more details refer the following links:- https://zenodo.org/record/1288822
For the results among 11112 to 30000 see the following link: https://zenodo.org/record/1998118
More work from Anne E. Bras and V.H.J. van der Velden in different bases:
– https://zenodo.org/record/2530974 (= base 11, 0 up to AAAA, incomplete)
– https://zenodo.org/record/2530976 (= base 12, 0 up to BBBB, incomplete)
– https://zenodo.org/record/2530978 (= base 13, 0 up to CCCC, incomplete)
– https://zenodo.org/record/2530980 (= base 14, 0 up to DDDD, incomplete)
– https://zenodo.org/record/2530982 (= base 15, 0 up to EEEE, incomplete)
– https://zenodo.org/record/2530984 (= base 16, 0 up to FFFF, complete)
– https://zenodo.org/record/2536047 (= base 16, -FFFFF up to FFFFF, complete)
– https://zenodo.org/record/2531777 (= base 11 up to 15, now all completed)