I am looking for all two consecutive integers A and A+1,which can be represented as sums of two squares $A=a^2+b^2$ and $A+1=c^2+d^2$, $a,b,c,d>0$
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Not a complete answer --- I doubt there is one --- but $$(n^2-n)^2+(n^2-n)^2,(n^2-2n)^2+(n^2-1)^2,(n^2-n-1)^2+(n^2-n+1)^2$$ gives three consecutive numbers, each a sum of two non-zero squares. This example is taken from Cochrane and Dressler, Consecutive triples of sums of two squares, which also cites earlier results about consecutive pairs of sums of two squares.