Going into more detail from RDKGame's reply, take the integer as n and the next as n+1 (as RDKGames says). Then square each so you have n^2 and (n+1)^2. Then add those two terms so you get n^2 + (n+1)^2 which = n^2 + n^2 + n + n + 1 = 2n^2 + 2n + 1 = 2(n^2 + n) + 1
Multiples of 2 are always even so 2(n^2 + n) must be even
An even number plus an odd number must be odd so 2(n^2 + n) + 1 must be odd.