A palindrome is usually a word (or sentence) which reads the same whether read forwards or backwards: for example "Madam, I'm Adam" (or her reply: "Eve").

We can also have palindromic numbers, such as 121, or 14641.

The prime numbers always provide a rich field for research. There are palindromic primes in base ten, and in bases other than ten, and some of these palindromic primes are palindromic in more than one base.

A good example (the most prolific so far) is 373; it has the forms 565 in base eight; 11311 in base four, and 454 in base nine. (I haven't checked 373 in bases above base ten yet)

Here is a list of results so far

Base | |

Three | 111, 212 (111 is also 11 base twelve) |

Four | 101, 131, 323 |

Five | 111,131, 232, 313, 414 |

(232 = 151 base six)(414=131 base nine) | |

Six | 11, 101, 111, 141, 151, 515, 525 |

(515 is 191 base ten, and 232 base nine) | |

Seven | 131, 212, 313, 515, 535, 616 |

(313 = 111 twelve) (515 = 10001 base four) | |

Eight | 111,131,141,161,323,343, 373, 535, 565, 717, 737, 747, 767 |

(343 = 272 base nine) | |

Nine | 131,151, 212, 232, 272, 414, 454, 515, 737, 757, 838, 858, 878 |

(858 = 10301 base five) (232 = 191 base ten) | |

(454 is 373 base ten) (212 = 20102 base three) |