# Rare numbers

Rare numbers are the non palindromic numbers^{} when added or subtracted to its reverse gives a perfect square^{}. For example 65 is a rare number because 65 + 56 = 11¡sup¿2¡/sup¿ and 65 - 56 = 3¡sup¿2¡/sup¿. Other examples of rare numbers are 65, 621770, 281089082, 2022652202, 868591084757, 872546974178 … (Sequence A035519 of OEIS). If we consider palindromic rare numbers, there are infinitely many rare numbers. For example, the numbers in the series 242, 20402, 2004002, 200040002, 20000400002 … are palindromic rare numbers. There are 84 rare numbers less than 10¡sup¿20¡/sup¿.

Rare numbers exhibit certain properties. Rare numbers start always with an even digit and end in the digits 0, 2, 3, 7 and 8. Digital root of a rare number is always 2, 5, 8 and 9. Odd rare numbers are much fewer than even rare numbers. Also rare numbers with odd number^{} of digits is fewer than rare numbers with even number of digits. After investigating rare numbers upto 10¡sup¿20¡/sup¿, Shyam Sunder Gupta conjectured that there are no prime numbers^{}. Also it is not known whether there are infinitely many rare numbers.

Title | Rare numbers |
---|---|

Canonical name | RareNumbers |

Date of creation | 2013-03-22 19:20:33 |

Last modified on | 2013-03-22 19:20:33 |

Owner | Kausthub (26471) |

Last modified by | Kausthub (26471) |

Numerical id | 4 |

Author | Kausthub (26471) |

Entry type | Definition |

Classification | msc 11A25 |

Defines | Numbers when added or subtracted to its reverse gives a perfect square |