Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 174–186

# An additive equation involving fractional powers

Article

## 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

11P55 11P32

## 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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