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Monday, January 7, 2019

560106 and 601065

There is a remarkable property of the sequence of numbers 560106, 5606106, 56066106, 560666106, etc. Take any number of the sequence, say 56066106, reverse the digits: 60166560, multiply these 2 numbers, multiply the result by 10 and the result is a perfect square. Thus
56066106×60166065×10=33732769778928900=1836648302.

To see this, let us denote the decimal digits reversal of a number n as R(n).   Let a=56×104+k+106+6000×(10k1)/9 for k0. Then R(a)=601×103+k+65+6000×(10k1)/9. The number 10×a×R(a) can be written as 30360100×(10k+31)2/9 whose square root is 5510×(10k+31)/3.

It is clear the the digit reversals of these numbers, i.e. 601065, 6016065, 60166065, 601666065, ..., satisfy the same property.

Other numbers with this property can be found in OEIS sequence A323061.

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