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Numbers  WON plate
192 |

[ January 15, 2015 ]
n and n_reversed are square numbers
whose roots may also be [ pseudo-] reversible
Andres Molina from Tulsa (email)

1089 is a smaller square number example whereby its reverse is also a square number
(n being not palindromic and without ending with zero).

12303690084 is the smallest square number whose reversal is also a square number
but so that their roots are pseudo-reversals amongst themselves (in base 10 at least).

What are the methods to transform nonreversible pairs into genuine reversibles ?
One way is to replace substrings using value n = – 1 like follows :

09 (leading zero included) can be replaced by 1n : which is in fact 10 – 1
099 (leading zero included) can be replaced by 10n : which is in fact 100 – 1
19 can be replaced by 2n : which is in fact 20 – 1
199 can be replaced by 20n : which is in fact 200 – 1
etc.

Let me clarify with next pair of reversible squares 12303690084 & 48009630321
110922 = 111n22
and
219111 = 22n111
As you can see now the roots 111n22 and 22n111 have changed into reversibles.

Another larger example with 102030003060900000804 & 408000009060300030201
10100990202 = 101010n0202
and
20199010101 = 2020n010101
And hey presto the reversibles 101010n0202 and 2020n010101 pop up.

Can you find more such number pairs whose square roots are [pseudo-] reversals as well ?
Or larger sporadic numbers beyond those listed in A061457 and A156316 ? When n and R(n) are equal they are of course palindromic. Square Root n(n)Reversal(n)Square Root R(n)
[Pseudo]Reversal
of Square Root n
122144441212
132169961312
33210899801992
102210404404012012
103210609906013012
GAP
31682100362244226300165012
205082420578064460875024214682
1109222
= 111n222
12303690084480096303212191112
= 22n1112
303577292158994929929499851293048772
11009222
= 1101n222
1212029250084480052920212121910112
= 22n10112
11092112
= 111n2112
12303490425211252409430321
°° root is palindromic !
11191112 °°
= 112n1112
11109222
= 1111n222
1234147690084480096741432121911112
= 22n11112
308036729488660854689986458066884931407932
101109222
= 10111n222
102230743690084480096347032201219111012
= 22n111012
110091112
= 1101n1112
121200525010321123010525002121110910112
= 111n10112
110091222
= 1101n1222
121200767210884488012767002121220910112
= 221n10112
110092212
= 1101n2212
121202947026841148620749202121121910112
= 122n10112
110191112
= 1102n1112
121420807230321123032708024121110920112
= 111n20112
110910222
= 111n10222
123010769004484484400967010321220091112
= 2201n1112
110911112
= 111n11112
123012743214321123412347210321111091112
= 1111n1112
110911212
= 111n11212
123012965036641146630569210321121091112
= 1211n1112
110911222
= 111n11222
123012987218884488812789210321221091112
= 2211n1112
110912022
= 111n12022
123014761804804408408167410321202091112
= 2021n1112
110912112
= 111n12112
123014961446521125644169410321112091112
= 1121n1112
110912122
= 111n12122
123014983628944449826389410321212091112
= 2121n1112
110921112
= 111n21112
123034926436321123634629430321111191112
= 1112n1112
111009222
= 11101n222
123230469250084480052964032321219101112
= 22n101112
111092112
= 1111n2112
123414569042521125240965414321111911112
= 112n11112
111109222
= 11111n222
123452587690084480096785254321219111112
= 22n111112
260490132678551078274169961472870155876310076262
31955891210211789696038811883069698711201433943512
101009902022
= 101010n02022
102030003060900000804408000009060300030201201990101012
= 2020n0101012
Related OEIS sequence
A115656 - Both n and the reverse of n are powerful(1) numbers (A001694).
A061457 - Numbers n such that n and its reversal are both squares.
A156316 - Perfect squares with property that their digit reversal is a larger perfect square.
A129914 -
Irregular square reversible numbers. Numbers which when squared and written
backwards give a square again, but don't satisfy reverse(n^2)=reverse(n)^2.
Index entry for sequences related to reversing digits of squares A000192 Prime Curios! Prime Puzzle  Wikipedia 192 Le nombre 192 ```

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