A1 Refereed original research article in a scientific journal
Roth-type theorem for nonlinear equations in Piatetski-Shapiro primes
Authors: Ren, Xiumin; Sun, Yu-Chen; Zhang, Qingqing; Zhang, Rui
Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD
Publishing place: SINGAPORE
Publication year: 2024
Journal: International Journal of Number Theory
Journal name in source: INTERNATIONAL JOURNAL OF NUMBER THEORY
Journal acronym: INT J NUMBER THEORY
Number of pages: 16
ISSN: 1793-0421
eISSN: 1793-7310
DOI: https://doi.org/10.1142/S1793042125500447
Web address : https://doi.org/10.1142/S1793042125500447
We consider the nonlinear diophantine equation c(1)p(1)(d) + c(2)p(2)(d) + center dot center dot center dot + c(s)p(s)(d) = 0 with c(1),c(2),...,c(s) is an element of Z being non-zero and satisfying c(1) + c(2) + center dot center dot center dot + c(s) = 0. We show that for s >= 2(sic)d(2)/2(sic) + 1 and c is an element of (1, 1 + c(0)(d,s)), if the equation has only K-trivial solutions in a subset epsilon of Piatetski-Shapiro primes up to x corresponding to c, then |A|<< x(c)(1)/log x (log log log log x)(2-s/dc +epsilon).
Funding information in the publication:
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871307 and 12031008) and the National Key Research and Development Program of China (Grant No. 2021YFA1000700). The second author was supported by UTUGS funding, working in the Academy of Finland Project No. 333707
