# An additive equation involving fractional powers

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## Abstract

Let \(1 < c < \frac{14}{11}\) and \(N\) be a sufficiently large integer. We prove that almost all \(n\in (N, 2N]\) can be represented as \(n=[p_1^c]+[p_2^c]\), where \(p_1\), \(p_2\) are prime numbers and \([x]\) denotes the integer part of \(x\). Our method also yields an asymptotic formula for the number of representations of these \(n\). The range \(1 < c < \frac{14}{11}\) constitutes an extension of \(1 < c < \frac{17}{16}\) due to Laporta.

## Key words and phrases

Diophantine equation exponential sum prime## Mathematics Subject Classification

11P55 11P32## Preview

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## Notes

### Acknowledgements

I thank the referee for his time and comments. The author would like to express the most sincere gratitude to Professor Yingchun Cai for his valuable and constant encouragement.

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