[ 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 |
|---|
| 650232 – 1 | 4227990528 | 8250997224 | 908352 – 1 |
| 1898082 – 1 | 36027076863 | 36867072063 | 1920082 – 1 |
| 1899082 – 1 | 36065048463 | 36484056063 | 1910082 – 1 |
| 17951032 – 1 | 3222394780608 | 8060874932223 | 28391682 – 1 |
| 189799082 – 1 | 360236907688463 | 364886709632063 | 191020082 – 1 |
| 189898082 – 1 | 360612807876863 | 368678708216063 | 192010082 – 1 |
| 189899082 – 1 | 360616605848463 | 364848506616063 | 191010082 – 1 |
| 589153702 – 1 | 3471020822236899 | 9986322280201743 | 999315882 – 1 |
| 1915139462 – 1 | 36677591512490915 | 51909421519577663 | 2278363922 – 1 |
| From here on terms are beyond what you can filter out from https://oeis.org/A066619 |
| 18979899082 – 1 | 3602365690869848463 | 3648489680965632063 | 19101020082 – 1 |
| 18980999082 – 1 | 3602783260749608463 | 3648069470623872063 | 19099920082 – 1 |
| 18989799082 – 1 | 3606124690987688463 | 3648867890964216063 | 19102010082 – 1 |
| 18989898082 – 1 | 3606162290887876863 | 3686787880922616063 | 19201010082 – 1 |
| 18989899082 – 1 | 3606162670685848463 | 3648485860762616063 | 19101010082 – 1 |
| 18990998082 – 1 | 3606580080745636863 | 3686365470800856063 | 19199910082 – 1 |
| 18990999082 – 1 | 3606580460565608463 | 3648065650640856063 | 19099910082 – 1 |
| 18991999082 – 1 | 3606960290547208463 | 3648027450920696063 | 19099810082 – 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}m9908 and 19{10}m1008 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
A000191 