Let the two perfect squares be  and 









Factor 53 to find possible values.   Because 53 is prime, there are only two factors:  1 and 53.



x-y = 1

x+y = 53





Add both equations:

2x = 54

x = 27 



This implies y = 26  











Check:

729 - 676 = 53





Because we can only factor 53 into 1*53, there is only one pair of perfect squares that differ by 53.  



For example, if we try

x - y = 53

x + y = 1



Adding gives:

2x = 54   ==>  x = 27  and  y = -26



This also works, but the perfect squares are the same as before: 729 and 676

 

Therefore, there is only one pair of perfect squares (729 and 676) whose difference is 53. 



----

  If you try a different difference, say 

you will find that there are three sets of perfect squares that have a difference of 63  (64 and 1,  144 and 81, 1024 and 961) and these correspond to the factorizations of 63  (7 * 9,   3 * 21, and 1 * 63,  respectively).



    

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