而图G的连通分支数，本质上就是高斯整数环中，范数为平方数的元素生成的等价类的数量。（等价关系定义为：整数u~v当且仅当存在有限个范数平方高斯整数构成的链，能将u和v作为分量连接起来）
However, the problem itself is ill-defined. If the domain of definition is a finite interval, it is trivial since a half circle would have easily satisfied the condition.
It is more interesting to stipulate the domain of definition to be on the whole real axis. In that case, there is a unique class of solutions that is that of the constant functions.
You can do it without differentiability but only continuity. Here is the proposition.
Proposition: The graph of a non-affine continuous function defined on the whole real line contains 4 concyclic points.
Proof: Choose three distinct noncollinear points on the graph; they determine a circle. If the graph intersects this circle at at least four points, we are done. Otherwise, it meets it in exactly three points.
Now consider the continuous function that, at each horizontal coordinate, measures the squared distance from the graph point to the center of the circle minus the squared radius. Its zeros are exactly those three intersection points, and it is positive for large positive and negative horizontal values because the graph is unbounded in the horizontal direction. We just need to consider the parity of the zero crossings. One of the three zeros is tangential. If the tangential zero is the middle one, then a slightly smaller concentric circle cuts the graph in four points; if it is one of the outer two, then a slightly larger concentric circle cuts the graph in four points. Thus, some circle intersects the graph in four distinct points. Therefore, every continuous non-affine function has four distinct concyclic points. ”
QED
Remark:
“the graph is unbounded in the horizontal direction”也可更准确的表述为“as x→±∞, g(x)→+∞”
Note the key correspondence below：
A triple (a,b,c) is a Pythagorean triple if and only if the Gaussian integer（a+bi）has norm N(a+bi)=c^2.
This correspondence linksGaussian integer structure to Pythagorean analysis.
该命题的核心是微分拓扑中的相交性质：定义在ℝ上的非仿射连续嵌入，无法被紧致圆周这类闭子集“局部约束”，因此其像必然与某一圆周 S¹⊂ℝ² 交于至少4个点。
该结论可推广至更一般的情形：任意ℝ非仿射光滑嵌入，均存在某一圆周，使得二者的交点数不少于4。
（注：陈省身先生与陈维桓合著的《微分几何讲义》（北京大学出版社）中，就系统的讨论了。（专门讲过这个结论）有兴趣的读者应该可以懂笔者的意思。）
笔者通宵了5天花了40余页让女友了解：极限，级数，Cauchy列，距离空间的收敛（于品数分）；同时也为了“物尽其用”，写完了一本陈祖维的纸质讲义。但现有成果，还远未达成笔者预设的kpi。
