Prove that the sum of all integers in L(P) never equals the sum of all integers in L(Q)\L(P).

It's known that S(2n-1) = n².

The sum of numbers in L(Q)\L(P) is S(Q) - S(P). If

S(Q) - S(P) = S(P)

then

S(Q) = 2S(P)

which can't be because on the left there is a square of an integer whilst on the right the number is not a square unless sqrt(2) is rational.