On the almost universality of ⌊𝑥²/𝑎⌋+⌊𝑦²/𝑏⌋+⌊𝑧²/𝑐⌋

Wu, Hai-Liang; Ni, He-Xia; Pan, Hao

doi:10.1090/tran/8438

On the almost universality of $\lfloor x^2/a\rfloor +\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$
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Trans. Amer. Math. Soc. 374 (2021), 7925-7944 Request permission

Abstract:

In 2013, B. Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor +\lfloor y^2/m\rfloor +\lfloor z^2/m\rfloor$ with $x,y,z\in \mathbb {Z}$, where $\lfloor \cdot \rfloor$ denotes the floor function. Moreover, in 2015, Z.-W. Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor +\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$ with $x,y,z\in \mathbb {Z}$, where $a,b,c$ are positive integers and $(a,b,c)\neq (1,1,1),(2,2,2)$.

In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, B. Farhi’s conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ pairwise coprime, we also confirm Z.-W. Sun’s conjecture for every sufficiently large integer $n$.

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