Thread: 3 squares

Show that among any 3 consecutive integers, at least one is a sum of three integer squares.

my working:
to solve this, lets consider m,m+1,m+2
we only need to consider the case where m is congruent to 7 mod 8 right?

Integer as Sum of Three Squares

i still do not get it. so if n=4^x(8k+7), we need to show that n+1 or n+2 is not of that form. how do we show that?

i'm wondering from the link, when we consider n = 8k+7 then n is congruent to 7 mod 8 but if n =4^x(8k+7) then n is congruent to 0 mod 8 if x>3 right? yet why does it say that it cant be a sum of three squares if n = 8k+7 and n =4^x(8k+7)?

Thank you!