On integers with many representations as the sum of $k$th powers of primes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915496519008256 |
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| author | Aggarwal, Anay |
| author_facet | Aggarwal, Anay |
| contents | For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$,
$$\limsup_{n\to\infty}f_k(n)=\infty.$$
This positively answers a conjecture of Erdos, which asks if there are natural numbers $n$ with arbitrarily many distinct representations of the form $p_1^k+p_2^k+\dots+p_k^k$ for primes $p_1,p_2,\dots,p_k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On integers with many representations as the sum of $k$th powers of primes Aggarwal, Anay Number Theory Combinatorics For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively answers a conjecture of Erdos, which asks if there are natural numbers $n$ with arbitrarily many distinct representations of the form $p_1^k+p_2^k+\dots+p_k^k$ for primes $p_1,p_2,\dots,p_k$. |
| title | On integers with many representations as the sum of $k$th powers of primes |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2509.11558 |