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Waring's Theorem for Powers of Primes


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1 Government College, Hoshiarpur, India
     

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The ideal Waring's Theorem stales that every positive integer is a sum of I integral nth powers ≥ 0, where

I = [(3/2)n]+2n-2.

Chowla and the author have considered the corresponding problem for primes so far as the squares and cubes are concerned. It has been shown that all positive integers i ≤ 240,000 are sums of at the most eight squares of primes p ≥ 1. It has also been shown that all positive integers i ≤ 100,000; i ≠ 1301 can be expressed as the sums of at the most twelve cubes of primes; while 1301 requires 13 and no less. In fact it is noticed that integers greater than 8161 require 11 cubes of primes at the most.


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  • Waring's Theorem for Powers of Primes

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Authors

Hans Raj Gupta
Government College, Hoshiarpur, India

Abstract


The ideal Waring's Theorem stales that every positive integer is a sum of I integral nth powers ≥ 0, where

I = [(3/2)n]+2n-2.

Chowla and the author have considered the corresponding problem for primes so far as the squares and cubes are concerned. It has been shown that all positive integers i ≤ 240,000 are sums of at the most eight squares of primes p ≥ 1. It has also been shown that all positive integers i ≤ 100,000; i ≠ 1301 can be expressed as the sums of at the most twelve cubes of primes; while 1301 requires 13 and no less. In fact it is noticed that integers greater than 8161 require 11 cubes of primes at the most.