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On a Particular Representation of Integers as Sums of kth Powers
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Consider the following way of representation of any positive integer x in the form
x = xlk+xk2+...+xks, (1)
where x1, x2, ..., xs are integers given by
(x1+1)k > x ≥ x1k
(x2+1)k > x-x1k ≥ x2k (2)
................
.................
the process terminating with
(xs+1)k > x-x1k-xk2-........-xks-1 = xks.
It can easily be seen that there is one and only one way of representation of an integer x in this manner. The number of kth powers required in the representation of x in the above manner is clearly a function of x and k, and hence can be denoted by Sk(x).
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