We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for $a\in\mathbb{N}$. Using symbolic computation techniques, these results are extended here to arbitrary $a\in\mathbb{R}$. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for $a=1,2,3,4$, and briefly discuss the case $0<a<1$.

翻译：我们对单色通用三价价(x,y,x+ay) $(x,y,x+ay) 进行精确和急剧的下限计算,这些三价(美元) 的条目来自 $1,\ dots,n ⁇ $,但有两种不同颜色的颜色。我们以前只知道这种三价的无色公式,而只有美元。使用符号计算技术,这些结果扩大到任意的 $a\ in\ mathbb{R} 。此外,我们给出了单色三价最低数的精确公式,以1,2,3,4美元计算,并简短地讨论案例 $0<a<1美元。