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Research Article | Open Access

On pairs of equations with unequal powers of primes and powers of 2

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, No. 995, Shangchuan Road, Pudong New Area, Shanghai 201209, China
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Abstract

It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations, each containing two squares of primes, two cubes of primes, two biquadrates of primes, and 30 powers of 2. Moreover, we also proved that every sufficiently large even integer can be expressed as the sum of two squares of primes, two cubes of primes, two biquadrates of primes, and 14 powers of 2. These two theorems constitute improvements upon the previous results.

Keywords

Waring–Goldbach problemHardy–Littlewood methodpowers of 2additive number theoryLinnik problem
CLC number: 11P05, 11P55

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AIMS Mathematics
Volume 10 Issue 2,
February 2025
Pages 4153-4172
DOI: 10.3934/math.2025193

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Zhu L. On pairs of equations with unequal powers of primes and powers of 2. AIMS Mathematics, 2025, 10(2): 4153-4172. https://doi.org/10.3934/math.2025193

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Received: 22 November 2024
Revised: 12 February 2025
Accepted: 17 February 2025
Published: 15 February 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)