# examples of 1-automorphic numbers

Concerning ourselves only with searching for automorphic numbers $n$ in bases $$ (binary to hexadecimal) and the ranges given by the iterator $$, and limiting to $m=1$ we find the following results:

First, it is obvious that 1 is a 1-automorphic number regardless of the base.

For the range and limit specified, there are no other 1-automorphic numbers in binary through quinary, bases 7 through 9, 11, 13 and 16.

In base 6, there are 1, 3, 4, 9, 28, 81, 136, and it is easy to verify that $3_{6}{}^{2}={13}_{6}$, $4_{6}{}^{2}={24}_{6}$, $13_{6}{}^{2}={213}_{6}$, $44_{6}{}^{2}={3344}_{6}$, etc.

In base 10, these ought to look familiar: 1, 5, 6, 25, 76, 376, 625.

Duodecimal: 1, 4, 9, 64, 81, 513, 1216. Noticing that 4 also appears in the list for base 6, we might wonder if 4 is always 1-automorphic when $6|b$? The question is moot because the next multiple^{} of 6 is $18>{4}^{2}$, thus in base 18 and any other higher bases, the square of 4 is also a 1-digit number.

Base 14: 1, 7, 8, 49, 148, 344, 2401.

Base 15: 1, 6, 10, 100, 126, 1000, 2376. Base 15 is the smallest odd base $b$ to have 1-automorphic numbers in the range specified. This should not be taken to imply that it is the smallest odd base to have automorphic numbers at all.

Title | examples of 1-automorphic numbers |
---|---|

Canonical name | ExamplesOf1automorphicNumbers |

Date of creation | 2013-03-22 16:20:20 |

Last modified on | 2013-03-22 16:20:20 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Example |

Classification | msc 11A63 |