Consider these two squares - 531441 and 254016
On the face of it, they appear like any other normal square. Let's look at some surprising facts
For starters 531441 is 729 * 729 and 254016 is 504 * 504 - Nothing unique as of now
a) Now look at the last 3 digits of both squares - 441 and 016 and both are perfect squares
b) Last digit of the square roots (729 and 504) - 9 and 4 and both are perfect squares
c) Difference between the square roots (729 and 504) is a perfect square again
d) 729 by itself is a perfect square as well as a cube
e) 504 is a unique number as it is the lowest three digit to have all factors from 1 to 9 except 5 (Once we multiple this number by 5 we get 2520 which is the smallest 4 digit number to have factors from 1 to 10)
f) Now dissect the original squares as following - 53 + 14 + 41 which gives 108 while 25 +40 + 16 gives 81 (108 and 81 are successive multiples of 27 which in turn is square root of 729!)
g) Multiple the non zero digits of both squares - 5*3*1*4*4*1 = 240, 2*5*4*1*6 = 240 again!
h) Prefix 1 to the square root of 531441 (which is 729) and we get the famous Ramanujan Number 1729. Now subtract the other square root 504 from 1729 and you get a perfect square 1225!
i) Suffix 1 to the square root of 254016 (which is 504) and we get a perfect square 5041 again!
j) Add 531441 and 254016 and the resultant number has factors 3,3,3,3 and a prime number 9697
k) Add 729 and 504 and the resultant number has factors 3,3 and a prime number 137! (see the pattern)
l) Subtract 531441 and 254016 and the resultant number has factors 3,3,3,3,5,5 and 137 again!
On the face of it, they appear like any other normal square. Let's look at some surprising facts
For starters 531441 is 729 * 729 and 254016 is 504 * 504 - Nothing unique as of now
a) Now look at the last 3 digits of both squares - 441 and 016 and both are perfect squares
b) Last digit of the square roots (729 and 504) - 9 and 4 and both are perfect squares
c) Difference between the square roots (729 and 504) is a perfect square again
d) 729 by itself is a perfect square as well as a cube
e) 504 is a unique number as it is the lowest three digit to have all factors from 1 to 9 except 5 (Once we multiple this number by 5 we get 2520 which is the smallest 4 digit number to have factors from 1 to 10)
f) Now dissect the original squares as following - 53 + 14 + 41 which gives 108 while 25 +40 + 16 gives 81 (108 and 81 are successive multiples of 27 which in turn is square root of 729!)
g) Multiple the non zero digits of both squares - 5*3*1*4*4*1 = 240, 2*5*4*1*6 = 240 again!
h) Prefix 1 to the square root of 531441 (which is 729) and we get the famous Ramanujan Number 1729. Now subtract the other square root 504 from 1729 and you get a perfect square 1225!
i) Suffix 1 to the square root of 254016 (which is 504) and we get a perfect square 5041 again!
j) Add 531441 and 254016 and the resultant number has factors 3,3,3,3 and a prime number 9697
k) Add 729 and 504 and the resultant number has factors 3,3 and a prime number 137! (see the pattern)
l) Subtract 531441 and 254016 and the resultant number has factors 3,3,3,3,5,5 and 137 again!
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