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

Li Zhu; Li Zhu

doi:10.3934/math.2025193

AIMS Mathematics

2025, Volume 10, Issue 2: 4153-4172. doi: 10.3934/math.2025193

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

Li Zhu ^,

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

Received: 22 November 2024 Revised: 12 February 2025 Accepted: 17 February 2025 Published: 27 February 2025
MSC : 11P05, 11P55

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.
- Waring–Goldbach problem,
- Hardy–Littlewood method,
- powers of 2,
- additive number theory,
- Linnik problem
Citation: Li Zhu. On pairs of equations with unequal powers of primes and powers of 2[J]. AIMS Mathematics, 2025, 10(2): 4153-4172. doi: 10.3934/math.2025193

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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.

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