Trevor Green claimed that:

**Theorem:** If Y is not in the range of f, and X does not end in 1, then XY is not in the range of f.

**Theorem:** If X is not in the range of f, and does not end in 1, then XY is not in the range of f.

and wondered whether:

**Conjecture:** If X is not in the range of f, then X has a substring of length 3 that is not in the range of f.

David Wilson noted that a finite state machine can be constructed which accepts the string representations of the elements in the range of f, or in the complement of the range of f, and that this still holds true in any base, and if any number of consecutive digits are added.

99(a+1) → 181a → 99(a+1) [where 0≤a≤8]

3333(a+1) → 666(a+4) → 12121a → 3333(a+1) [where 0≤a≤5]

Philippe Fondanaiche found some of these. Berend Jan van der Zwaag found that 733 numbers end in the first cycle and 222 numbers end in the second cycle.

Trevor Green conjectured that these are the only periodic points in base 10. He also made the observation that every periodic cycle includes a number that starts with 1.

Joseph DeVincentis and Berend Jan van der Zwaag gave the largest number containing no zeroes that converges to 0 (831112), and Berend Jan van der Zwaag gave the largest number that does not contain 2 consecutive zeroes which conveges to 0 (8010120). Joseph DeVincentis gave a short proof that every number smaller than 991 converges to 0.

Trevor Green conjectured that there are only finitely many numbers in any base that go to 0 besides the numbers a0...0b, which he calls *bodiless* numbers. Berend Jan van der Zwaag confirmed this, and gave the complete list. Trevor Green gives the following list:

base | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|

n | 10^{2}+10 | 2×10^{5}+10 | 3×10^{7}+10 | 10^{10}+10 | 2×10^{10}+10 | 4×10^{12}+10
| 7×10^{13}+10 | 3×10^{15}+10 | 9×10^{16}+10 | 4×10^{18}+10 |

Berend Jan van der Zwaag claims the largest number with 3 non-zero digits which converges to 0 is 11×10^{15}+8.

Brendan Owen and Berend Jan van der Zwaag gave a list of the smallest n for which f^{k}(n)=0:

k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | 0 | 1 | 10 | 19 | 109 | 149 | 197 | 399 | 694 | 796 | 893 | 897 | 1167 | 1579 | 1596 | 1667 | 1790 | 1777 | 2859 | 1779 | 1778 | 1873 | 3679 | 5926 | 11289 | 9539 |

k | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | >40 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | 13551 | 4589 | 5960 | 12066 | 12265 | 19119 | 10927 | 12379 | 11742 | 65220 | 34038 | 40390 | 1110025 | 10100023 | 91000021 | 1000...0009 |

Berend Jan van der Zwaag confirmed this for some small cases. When adding 3 consecutive digits, numbers smaller than 10889 converge to 0, 10889 converges to ∞, and the smallest periodic cycle is 125610 → 813127 → 125610. When adding 4 consecutive digits, numbers smaller than 1005952 converge to 0 except for the few fixed point mentioned below, 1005952 converges to ∞, and the smallest fixed points are 18181a. He also found some periodic points:

8888888(a+8) → 323232323(a+2) → 1010101010101a → 2222222222(a+2) → 8888888(a+8)

666666(a+4) → 2424242(a+2) → 121212121a → 666666(a+4).

He used this observation to completely analyze the binary case. He proved that 1000...0001, 1000...000, 11, 1, and 110 go to 0, 111, 1010, and 1100 end up in the 111 → 1010 → 111 loop, and every other number diverges!

Brendan Owen gave a list of periodic points in bases no more than 10. In every base b, there are (b–1) 2-cycles of the form (b–1)(b–1)c → 1(b–2)1(c–1) → (b–1)(b–1)c, where 1≤c≤b–1. These also appear to be the only 2-cycles. Here is the list of the other known periodic points in other small bases:

Base | Larger Cycles |
---|---|

4 | (2222,101010,11111), (2223,101011,11112) |

5 | (2331,10114,11210) |

7 | (4443,111110,22221), (4444,111111,22222), (4445,111112,22223), (4446,111113,22224) |

8 | (10101010,1111111,222222,44444), (10101011,1111112,222223,44445), (10101012,1111113,222224,44446), (10101013,1111114,222225,44447) |

9 | (6627,113810,412101,53311,8642,15116), (6628,13811,412102,53312,8643,15117) |

10 | (33331,6664,121210), (33332,6665,121211), (33333,6666,121212), (33334,6667,121213), (33335,6668,121214), (33336,6669,121215) |

Trevor Green found that if a and n are given, the number consisting of 2n+1 digit a's has period n in base a(2^{n}–1) + 1. He points out that base 16 has 3 of these cycles, base 64 has 4, and base 316 has 5. He asks whether all cycles which are not 2-cycles are of this form, but doubts the answer is yes.

In 2011, Lars Blomberg sent me this information about this problem.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/27/11.