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EXAMPLE
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12 and 21 are rotationally connected and also their squares 144, 441 are obtained from each other by rotation of their decimal representations.
Also 122 and 221 are rotationally connected as well as their squares 14884 and 48841.
Notice infinite pattern (n,m)= (12...2, 2...21).
Corresponding squares:
{144, 441},
{14884, 48841},
{1493284, 4932841},
{21298225, 29822521},
{149377284, 493772841},
{14938217284, 49382172841},
{161640986116, 409861161616},
{363692218761, 922187613636},
{755232259681, 817552322596},
{1493826617284, 4938266172841}.
There is another infinite subsequence:
The cyclic pair (201023, 320102)*k (for k = 2 and 3) and its squares (40410256529, 102465290404)*k^2.
The next in the sequence is:
(020001000203, 30200010002)*k,(400040009120406041209, 912040604120900040004)*k^2 (for k = 16, 17, ..., 33).
In general: (0{n}20{2n+1}10{2n+1}20{n}3, 30{n}20{2n+1}10{2n+1}20{n}3)* k, where lower bound of k = 5*10^(n-1)*sqrt(10) and upper bound of k = 3{n+1} for n = 0, 1, 2, 3, 4, etc.
For example, n = 6 gives lower bound k = 1581139 with lower cyclic pair: (316227800000001581139000000031622784743417, 474341731622780000000158113900000003162278)
and corresponding squares:(100000021492841000000214928422500007835889744785236492844223926514928486978524835889, 225000078358897447852364928442239265149284869785248358891000000214928410000002149284)
and upper bound k = 3333333 with upper cyclic pair:(666666600000003333333000000066666669999999, 999999966666660000000333333300000006666666)
and its corresponding squares:
(444444355555564444443555555699999993333332111111222222217777778888888766666660000001,
999999933333321111112222222177777788888887666666600000014444443555555644444435555556).
(End)
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