Consecutive square-free values of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$

Feng, Ya-Fang. "Consecutive square-free values of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$." Czechoslovak Mathematical Journal 73.1 (2023): 297-310. <http://eudml.org/doc/299442>.

@article{Feng2023,
abstract = {We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^\{2\}+y^\{2\}+z^\{2\}+k$, $x^\{2\}+y^\{2\}+z^\{2\}+k+1$. We also establish an asymptotic formula for $1\le x, y, z \le H$ such that $x^\{2\}+y^\{2\}+z^\{2\}+k$, $x^\{2\}+y^\{2\}+z^\{2\}+k+1$ are square-free. The method we used in this paper is due to Tolev.},
author = {Feng, Ya-Fang},
journal = {Czechoslovak Mathematical Journal},
keywords = {square-free number; Salié sum; Gauss sum},
language = {eng},
number = {1},
pages = {297-310},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Consecutive square-free values of the type $x^\{2\}+y^\{2\}+z^\{2\}+k$, $x^\{2\}+y^\{2\}+z^\{2\}+k+1$},
url = {http://eudml.org/doc/299442},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Feng, Ya-Fang
TI - Consecutive square-free values of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 297
EP - 310
AB - We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$. We also establish an asymptotic formula for $1\le x, y, z \le H$ such that $x^{2}+y^{2}+z^{2}+k$, $x^{2}+y^{2}+z^{2}+k+1$ are square-free. The method we used in this paper is due to Tolev.
LA - eng
KW - square-free number; Salié sum; Gauss sum
UR - http://eudml.org/doc/299442
ER -