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March 2021 Small prime solutions of a Diophantine equation with one prime and five cubes of primes
Weiping Li
Funct. Approx. Comment. Math. 64(1): 47-69 (March 2021). DOI: 10.7169/facm/1874

## Abstract

Let $a_1,\cdots,a_6$ be non-zero integers satisfying $(a_i,a_j)=1, 1\leq i \lt j \leq 6$ and $b$ be any integer. For the Diophantine equation $a_1p_1+a_2p_2^3+\cdots+a_6p_6^3=b$ we prove that (i) if all $a_1,\cdots,a_6$ are positive and $b\gg \max \{|a_j|\}^{34+\varepsilon}$, then the equation is soluble in primes $p_j$, and (ii) if $a_1,\cdots,a_6$ are not all of the same sign, then the equation has prime solutions satisfying $\max \{ p_1,p_2^3,\cdots,p_6^3 \}\ll |b|+\max \{|a_j|\}^{33+\varepsilon}$, where the implied constants depend only on $\varepsilon$.

## Citation

Weiping Li. "Small prime solutions of a Diophantine equation with one prime and five cubes of primes." Funct. Approx. Comment. Math. 64 (1) 47 - 69, March 2021. https://doi.org/10.7169/facm/1874

## Information

Received: 5 March 2020; Revised: 16 June 2020; Published: March 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.7169/facm/1874

Subjects:
Primary: 11P05 , 11P32 , 11P55

Keywords: circle method , Diophantine equation , small prime solution  