O n observing the triple (25, 125, 225) in which 125 is a perfect cube, 25 and 225 are perfect squares, and the three numbers are in arithmetic progression (AP), I felt that 125 is a very special perfect cube which is guarded by two perfect squares on either side at equal distance.

A surprising discovery we make is that 125 is guarded by two perfect squares in another way, namely: (81, 125, 169); here, 81 and 169 are perfect squares, and the three numbers are in AP as earlier.

I wondered about the existence of other such perfect cubes. If they exist, then on what condition? If not, then why?

I named such triplets ‘CuRe Triplets’ (Cube-SquaRe).

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