"Can you think what is the similarity among the numbers 17689, 18769, 78961 and 81796? ". You will surprise to note that all have like digits (i.e. 1,6,7,8 and 9) and at the same time all are perfect squares i.e. the square of 133, 137, 281 and 286 respectively.

A systematic study of Square numbers up to 105 was made and reported by me in [1], which revealed that:

Some of the interesting observations of Squares having like digits as reported earlier by me in [1] are:

Recent Study:

I have now made a study of all Squares up to 1010 which reveals that up to 1010:

Curious observations:

Cubes and higher Powers:

Apart from squares, there are cubes and fourth powers also which have like digits. For example, there are 6 pairs of cubes up to 107 which have like digits. These are (125, 512), (42875, 54872), (125000, 512000), (1030301, 1331000), (1061208, 8120601) and (5639752, 7529536). It can be seen that there are two reversible pairs in the above examples.

Three pairs of fourth powers having like digits up to 107 are (256, 625), (1048576, 5764801) and (2560000, 6250000).

It is interesting to note that squares and cubes of pairs of numbers also have like digits. For example: (101, 110), (102, 201) and (178, 196).

It can be good past time for Number enthusiasts to find squares, cubes and higher powers having like digits for higher numbers.


[1] Squares, cubes and biquadrates having like digits, S. S. Gupta, Science Reporter, June 1989, India.
[2]Beiler, Albert H., Recreations in the Theory of Numbers, Dover, New York, 1966.
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