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On a Particular Representation of Integers as Sums of kth Powers


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1 Andhra University, India
     

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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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  • On a Particular Representation of Integers as Sums of kth Powers

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Authors

K. Sambasiva Rao
Andhra University, India

Abstract


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).