In the thesis, Schur's theorem on sum-free partitions is proven and Schur number S(n) is defined as the largest positive integer with the property that the set {1,...,S(n)} can be partitioned into n sum-free subsets. Values of known Schur numbers S(1) to S(5) are given as well as some upper and lower bounds for general S(n). Weak Schur numbers are also defined. Moreover, Schur's theorem is formulated as a graph coloring problem and presented as a corollary of Ramsey theorem. In conclusion, Schur's theorem is linked to Fermat's last theorem. Schur's simplification of Dickson's proof that equation x^n+y^n=z^n for fixed n > 2 has nontrivial solutions in Z_p for all sufficiently large prime p is given.
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