153 is an Armstrong number by definition and that's a given but there are some unique features that this number exhibits in and around its ecosystem
Here are few gems starting from some obvious ones to not so obvious
a) 153 - 9 is a perfect square (note 9 is also a perfect square)
b) 153 + 16 is a perfect square (note 16 is also a perfect square)
c) 153 - 72 is a perfect square (81)
d) 153 + 72 is a perfect square again (225)
e) 153 + 81 = 234 and 153-144 = 09 and when we concatenate 234 and 09 we get 23409 which is 153 * 153! (153 squared)
f) 153 is sum of two consecutive numbers 77 and 76 which is normal for any odd number but here is when it gets interesting - 7776 by itself is 6 ^ 5, 7776 + 8100 (which is a perfect square( is another perfect square (15876), 7776 + 7600 (00 concatenated with 76) gives 15376 which is not only a perfect square but it is our Armstrong number 153 combining with 76!
g) Continuing from the previous point, the beauty of numbers become evident when you try to combine 153 with 77 - 15377 is a prime number with no factors unlike 15376 which is a perfect square
h) 1 ^ 3 + 5 ^ 3 + 3 ^ 3 = 153 as we know. Now add 4 ^ 3 to that and we get 217. Now add 153 and 217 and we get 370 which by itself is another Armstrong number!
Here are few gems starting from some obvious ones to not so obvious
a) 153 - 9 is a perfect square (note 9 is also a perfect square)
b) 153 + 16 is a perfect square (note 16 is also a perfect square)
c) 153 - 72 is a perfect square (81)
d) 153 + 72 is a perfect square again (225)
e) 153 + 81 = 234 and 153-144 = 09 and when we concatenate 234 and 09 we get 23409 which is 153 * 153! (153 squared)
f) 153 is sum of two consecutive numbers 77 and 76 which is normal for any odd number but here is when it gets interesting - 7776 by itself is 6 ^ 5, 7776 + 8100 (which is a perfect square( is another perfect square (15876), 7776 + 7600 (00 concatenated with 76) gives 15376 which is not only a perfect square but it is our Armstrong number 153 combining with 76!
g) Continuing from the previous point, the beauty of numbers become evident when you try to combine 153 with 77 - 15377 is a prime number with no factors unlike 15376 which is a perfect square
h) 1 ^ 3 + 5 ^ 3 + 3 ^ 3 = 153 as we know. Now add 4 ^ 3 to that and we get 217. Now add 153 and 217 and we get 370 which by itself is another Armstrong number!
