On a Diophantine equation involving fractional powers with primes of special types

Liuying Wu; Liuying Wu

doi:10.3934/math.2024799

AIMS Mathematics

2024, Volume 9, Issue 6: 16486-16505. doi: 10.3934/math.2024799

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Research article

On a Diophantine equation involving fractional powers with primes of special types

Liuying Wu ^,

School of Mathematical Science, Tongji University, Shanghai 200092, China

Received: 23 March 2024 Revised: 22 April 2024 Accepted: 25 April 2024 Published: 13 May 2024
MSC : 11L07, 11L20, 11N35, 11N36

Suppose that $ N $ is a sufficiently large real number. In this paper it is proved that for $ 2 < c < \frac{990}{479} $, the Diophantine equation

$ \left[p_{1}^{c}\right]+\left[p_{2}^{c}\right]+\left[p_{3}^{c}\right]+\left[p_{4}^{c}\right]+\left[p_{5}^{c}\right] = N $

is solvable in primes $ p_{1}, p_{2}, p_{3}, p_4, p_5 $ such that each of the numbers $ p_{i}+2, i = 1, 2, 3, 4, 5 $ has at most $ \left[\frac{6227}{3960-1916c}\right] $ prime factors.
- almost prime,
- diophantine equality,
- fractional powers,
- exponential sum
Citation: Liuying Wu. On a Diophantine equation involving fractional powers with primes of special types[J]. AIMS Mathematics, 2024, 9(6): 16486-16505. doi: 10.3934/math.2024799

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Abstract

Suppose that $ N $ is a sufficiently large real number. In this paper it is proved that for $ 2 < c < \frac{990}{479} $, the Diophantine equation

$ \left[p_{1}^{c}\right]+\left[p_{2}^{c}\right]+\left[p_{3}^{c}\right]+\left[p_{4}^{c}\right]+\left[p_{5}^{c}\right] = N $

is solvable in primes $ p_{1}, p_{2}, p_{3}, p_4, p_5 $ such that each of the numbers $ p_{i}+2, i = 1, 2, 3, 4, 5 $ has at most $ \left[\frac{6227}{3960-1916c}\right] $ prime factors.

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