Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares.
First few numbers are 3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,⋯
Sums of squares can only be of the form 4k, 4k+1 and 4k+2. So can we use this idea to prove the proposition?
I didn't find a logical sequence. Can anyone provide some hints to proceed with this?
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