FASCINATING SQUARES |

*"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 10^{5} was
made and reported by me in [1], which revealed that:

- There is only one set of six squares i.e. (10609, 16900, 19600, 61009,
90601, 96100) and one set of five squares i.e. (16384, 31684, 36481, 38416,
43681) up to 10
^{5}having like digits. - There are two sets of four squares i.e. (10404, 14400, 40401, 44100) and
(17689, 18769, 78961, 81796) up to 10
^{5}having like digits. - There are nine sets of three squares i.e. (169,
196, 961), (1296, 2916, 9216), (11664, 16641, 41616), (12544, 44521, 52441),
(18225, 25281, 81225), (21904, 41209, 91204), (23716, 32761, 72361), (25600,
60025, 62500) and (42849, 49284, 82944) up to 10
^{5}having like digits. - Besides above there are number of pairs of squares having like digits. For
example 144 and 441, both are squares and have like digits. There are 30 pairs
of squares up to 10
^{5}having like digits excluding trivia's like (100, 001), (400,004) etc. These 30 pairs do not include the pairs, which can be formed from higher set of numbers mentioned above. - Following are the 30 pairs of squares having like digits: (144,441), (256,625), (1024, 2401), (1089,9801), (1369,1936), (1764, 4761), (4096,9604), (10201, 12100), (11236, 21316), (12769,96721), (14884, 48841), (20736,30276), (23104, 32041), (23409, 39204), (24025, 42025), (24649,24964), (27556, 75625), (29584, 54289), (34596, 45369), (36100, 63001), (40804, 48400), (36864, 86436), (42436, 43264), (46656, 66564), (50176, 51076), (50625, 65025), (51984, 95481), (70756, 75076), (74529, 79524) and (98596, 99856).

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

- There are pairs of consecutive numbers like (13, 14) and (157, 158), whose squares have like digits i.e. (169, 196) and (24649, 24964) respectively.
- There are pairs of consecutive even numbers like (206, 208), (224, 226) and (314, 316), whose squares have like digits i.e. (42436, 43264), (50176, 51076) and (98596, 99856) respectively.
- There are pairs of reversible numbers like (12, 21), (13, 31), (112, 211), (102, 201), (103, 301), (122, 221) and (113, 311), whose squares have like digits.

**Recent Study:**

I have now made a study of all Squares up to 10^{10}
which reveals that up to 10^{10}:

- There are 7553 pairs of squares having like digit.
- There are 4119 triplets of squares having like digit.
- There is one set of 87 squares having like digit. All squares in this set contain all ten digits from 0 to 9 (unrepeated). This has earlier been mentioned in [2]. For all these 87 numbers, visit the great site of World of Numbers by Patrick De Geest
- There is one set of 46 squares having like digits. This is given by the square of the following numbers: 32164, 35494, 38315, 38536, 39067, 40385, 42703, 43985, 45754, 46136, 46258, 46267, 46433, 47572, 47801, 48013, 48445, 49853, 50209, 50839, 50906, 51073, 51221, 51302, 53116, 54169, 57013, 57395, 59852, 60248, 62471, 62899, 62935, 66301, 68105, 72394, 72943, 72947, 78347, 82595, 90205, 91429, 91685, 95825, 96145 and 99205.
- There is one set of 39 squares, 2 sets of 36 squares, 2 sets of 33 squares, 4 sets of 32 squares, 7 sets of 31 squares having like digits.
- There are 6 sets of 30 squares having like digits. Out of these one set of 30 squares contains all nine digits from 1 to 9 (unrepeated). This set of 30 square numbers has been given in [2].
- There are 2 sets of 29 squares, 3 sets of 28 squares, 5 sets of 27 squares, 4 sets of 26 squares, 12 sets of 25 squares, 11 sets of 24 squares, 17 sets of 23 squares having like digits.
- There are 7 sets of 22 squares having like digits. Out of these one set of 22 squares contain all digits from 0 to 8 (unrepeated). This is given by the square of the following numbers: 10128, 10278, 12582, 13278, 13434, 13545, 13698, 14442, 14766, 16854, 17529, 17778, 20754, 21744, 21801, 23682, 23889, 24009, 27105, 27984, 28731 and 29208.
- There are 19 sets of 21 squares, 24 sets of 20 squares, 32 sets of 19 squares, 36 sets of 18 squares, 63 sets of 17 squares, 58 sets of 16 squares, 74 sets of 15 squares, 93 sets of 14 squares, 133 sets of 13 squares, 165 sets of 12 squares, 212 sets of 11 squares having like digits.
- There are 282 sets of 10 squares, 423 sets of 9 squares, 558 sets of 8 squares, 840 sets of 7 squares, 1094 sets of 6 squares, 1599 sets of 5 squares and 2602 sets of 4 squares having like digits.

**Curious observations**:

- There are some pairs of numbers containing all ten digits from 0 to 9 (unrepeated) and each gives squares containing all ten digits unrepeated. For example the pair (35172, 60984) contain all ten digits and square of each number of pair i.e. (1237069584, 3719048256) also contain all ten digits unrepeated. Other example are (57321, 60984), (58413, 96702) and (59403, 76182).
- There are some pairs and triplets of numbers having like digits and whose squares contain all ten digits unrepeated. For example (35172, 57321), (78453, 85743), (58455, 58554) and (54918, 81945, 89145).
- There are only 3 palindromes whose square contains all ten digits unrepeated. These are 35853, 84648 and 97779.This has been mentioned earlier on the web site of Patrick De Geest
- There is no palindromic number whose square contains all nine digits (i.e. 1 to 9) unrepeated.

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
10^{7} 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 10^{7
}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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