[ *January 4, 2015* ]

n and its reverse are one less than a square number

Andres Molina from Tulsa (email)

Andres asks for more reversal pairs with this property.

Both n and its reverse are one less than a square number

(n being non-palindromic and without ending with zero).

4227990528 is the smallest number one less than a square number

and reverse it that is one less than a square number that is not

palindromic and without ending with zero in base 10.

Square Root n – 1 | (n) | Reversal(n) | Square Root R(n) – 1 |

65023^{2} – 1 | 4227990528 | 8250997224 | 90835^{2} – 1 |

189808^{2} – 1 | 36027076863 | 36867072063 | 192008^{2} – 1 |

189908^{2} – 1 | 36065048463 | 36484056063 | 191008^{2} – 1 |

1795103^{2} – 1 | 3222394780608 | 8060874932223 | 2839168^{2} – 1 |

18979908^{2} – 1 | 360236907688463 | 364886709632063 | 19102008^{2} – 1 |

18989808^{2} – 1 | 360612807876863 | 368678708216063 | 19201008^{2} – 1 |

18989908^{2} – 1 | 360616605848463 | 364848506616063 | 19101008^{2} – 1 |

58915370^{2} – 1 | 3471020822236899 | 9986322280201743 | 99931588^{2} – 1 |

191513946^{2} – 1 | 36677591512490915 | 51909421519577663 | 227836392^{2} – 1 |

From here on terms are beyond what you can filter out from https://oeis.org/A066619 |

1897989908^{2} – 1 | 3602365690869848463 | 3648489680965632063 | 1910102008^{2} – 1 |

1898099908^{2} – 1 | 3602783260749608463 | 3648069470623872063 | 1909992008^{2} – 1 |

1898979908^{2} – 1 | 3606124690987688463 | 3648867890964216063 | 1910201008^{2} – 1 |

1898989808^{2} – 1 | 3606162290887876863 | 3686787880922616063 | 1920101008^{2} – 1 |

1898989908^{2} – 1 | 3606162670685848463 | 3648485860762616063 | 1910101008^{2} – 1 |

1899099808^{2} – 1 | 3606580080745636863 | 3686365470800856063 | 1919991008^{2} – 1 |

1899099908^{2} – 1 | 3606580460565608463 | 3648065650640856063 | 1909991008^{2} – 1 |

1899199908^{2} – 1 | 3606960290547208463 | 3648027450920696063 | 1909981008^{2} – 1 |

After having a look at the square roots one can easily detect some nice patterns

18|9908

18989908

1898989808

189898989908

etc.

19|1008

1910101008

191010101008

19101010101008

etc.

Highlighted in color are the inserted substrings. In the first case it is 98

In the second case it is 10.

Let us write this pattern in a more mathematical manner.

18{98}_{m}9908 and 19{10}_{m}1008 both with m = 0, 1, 2, 3, 4

giving birth to following five number pairs n and R(n)

36065048463 | 36484056063

360616605848463 | 364848506616063

3606162670685848463 | 3648485860762616063

36061626368078685848463 | 36484858687086362616063

360616263646908878685848463 | 364848586878809646362616063

The pattern stops here and is alas not infinite.

A carry occured somewhere in the middle of n making it no longer reversable.

Can other analogue patterns be detected as well, finite or even better __infinite__ ?

When n and R(n) are equal they are of course palindromic.

The table with these solutions was constructed a couple of years ago.

Please link to Palindromic Quasipronic Numbers of the form n(n+2)" which is also n^2–1