Waring's Problem for N = 2
نویسنده
چکیده
Given a natural number n, Waring’s problem asks for the minimum natural number sn such that every natural number can be represented as the sum of sn nth powers of integers. In this paper, we will answer this question for the case n = 2. To do this, we will examine the properties of two rings, one of which is a subring of the complex numbers, the other of which is a subring of the quaternions. These have naturally defined norms and division algorithms, which we will use to prove the result that all numbers can be written as the sum of four squares, as well as giving a necessary and sufficient condition for the sum of two squares.
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