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Problem 604

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Convex path in square

Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass through.

You are given that $F(1)=2$, $F(3)=3$, $F(9)=6$, $F(11)=7$, $F(100)=30$ and $F(50000)=1898$.
Below is the graph of a function reaching the maximum 3 for $N=3$:

Find $F({10}^{18})$.

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正方形中的凹路径

记$F(N)$为，在一个和坐标系对齐的$N\times N$正方形中，一个严格凹单调增函数所能够穿过的格点数目最大值。

已知$F(1)=2$，$F(3)=3$，$F(9)=6$，$F(11)=7$，$F(100)=30$以及$F(50000)=1898$。
以下是$N=3$时达成最大值的函数图象：

求$F({10}^{18})$。

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