Lagrange's four square theorem has been generalized in many directions. This paper presents two generalizations of Lagrange’s four-square theorem. 

First, we investigate the minimum number of squares required to represent any non-negative integer without using a fixed positive integer $\rho$. we define $M(\rho)$ the smallest integer $k$ such that every positive integer can be represented as a sum of at most $k$ squares, excluding a fixed integer $\rho$ from the set of summands. We determinded that $M(2)=8$, $M(1)=M(3)=6$, $M(\rho)=5$ if \ $\rho=5,\ 2^{m+1},\ 3\cdot2^m $ for some positive integer $m$, and $M(\rho)=4$ if otherwise. 

Second, we study the representability of non-negative integers as sums of four squares from ``thin" subsets $S \subset \mathbb{Z}$. In particular, we prove that we can take $S$ to be a density zero set. We mainly use Hua's theorem and the sieve method.