On the Representation of Integers as Sums of Limited Prime Powers

Abstract

We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five prime powers p^k, where p is a prime number and k is an integer greater than or equal to 2. This conjecture is supported by comprehensive computational evidence covering all integers up to 10^7(exhaustively) and specific large numbers up to 10^10 (via sampling), where no counterexample requiring more than five summands has been found. This work highlights a surprising "combinatorial creativity" of prime powers, which allows for efficient additive representations despite their asymptotic sparsity and the existence of extremely large gaps between individual prime powers.