On the Representation of Integers as Sums of Limited Prime Powers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913974093611008 |
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| author | Stricker, Julius |
| author_facet | Stricker, Julius |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Representation of Integers as Sums of Limited Prime Powers Stricker, Julius General Mathematics 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. |
| title | On the Representation of Integers as Sums of Limited Prime Powers |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2508.01686 |