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prove that there is only one pair of (integer) perfect squares that differ by 53, and find it

 Jan 18, 2024
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prove that there is only one pair of (integer) perfect squares that differ by 53, and find it    

 

Let's call the squares x2 and y2 so                  y2 – x2  =  53   

 

Well, I couldn't figure out how to do it, so   

I went to Desmos and plotted the curve of       y  =  sqrt(x2 + 53)   

and by careful scrutinization, discovered  

the place where the curve crossed two  

whole numbers on the grid.  It wasn't    

as hard as describing it sounds.  

The answer is when x = 26 & y = 27               272 – 262  =  53   

 

                                                                       729676  =  53   

I can't prove that's the only pair.  Sorry.   

 

.

 Jan 18, 2024

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