Let $ p_{1}, p_{2}, \ldots, p_{7} $ be primes. In this paper, we first show that when $ k_{1} = 23 $, every sufficiently large odd integer can be represented as the sum of one prime square, five prime cubes, one prime biquadrate and at most $ k_{1} $ powers of $ 2 $. We further prove that for $ k_{2} = 45 $, every pair of sufficiently large odd integers satisfying certain necessary conditions can be represented as a pair of equations involving one prime square, five prime cubes, one prime biquadrate and at most $ k_{2} $ powers of $ 2 $.
Citation: Xue Han, Rui Zhang. Goldbach-Linnik type problems for mixed powers of primes and powers of 2[J]. Electronic Research Archive, 2025, 33(12): 7902-7917. doi: 10.3934/era.2025348
Let $ p_{1}, p_{2}, \ldots, p_{7} $ be primes. In this paper, we first show that when $ k_{1} = 23 $, every sufficiently large odd integer can be represented as the sum of one prime square, five prime cubes, one prime biquadrate and at most $ k_{1} $ powers of $ 2 $. We further prove that for $ k_{2} = 45 $, every pair of sufficiently large odd integers satisfying certain necessary conditions can be represented as a pair of equations involving one prime square, five prime cubes, one prime biquadrate and at most $ k_{2} $ powers of $ 2 $.
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Xue Han, Rui Zhang. Goldbach-Linnik type problems for mixed powers of primes and powers of 2[J]. Electronic Research Archive, 2025, 33(12): 7902-7917. doi: 10.3934/era.2025348
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