Prove (2^(2n+1))+2 is equal to the sum of two different squares

[Seabo04]

Prove (2^(2n+1))+2 is equal to the sum of two different squares
N is a positive integer

 

[Romsek]

Have you tried induction?

 

[Steven G]

Have you tried induction?

What do you think?

 

[Dr.Peterson]

Prove (2^(2n+1))+2 is equal to the sum of two different squares
N is a positive integer

I tried a few examples until I found a pattern in the squares to be summed; then it was easy to prove my conjecture algebraically. But I'll admit I used a spreadsheet to find the pattern. In principle, you could find it by trying to rearrange the expression as a sum of squares.

 

[lookagain]

5

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Prove (2^(2n+1))+2 is equal to the sum of two different squares
N is a positive integer

A couple of observations:

For n = 1 to 3, one of the squares can be 9, but with n = 4, one of the squares is not 9.
With n = 3, there are two different sets of squares that work.