On integers with many representations as the sum of kth powers of primes

2025/09/15 by Anay Aggarwal, Aggarwal, Anay
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.11558

openalex publication_date 2025/09/15 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

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→∞f_k(n)=∞. This positively answers a conjecture of Erdos, which asks if there are natural numbers n with arbitrarily many distinct representations of the form p₁^k+p₂^k+…+p_k^k for primes p₁,p₂,…,p_k.

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