"The concept the problem is describing is called a [Farey sequence](https://en.wikipedia.org/wiki/Farey_sequence). The example given in the problem is $F_8$, and we are tasked with finding the numerator of the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
"It turns out there is a very simple method for determining this. Whenever you have two neighbors $\\frac{a}{b}$ and $\\frac{c}{d}$ in a Farey sequence, the next term that will appear between them in a subsequent Farey sequence is simply $\\frac{a+c}{b+d}$, called the [mediant](https://w.wiki/EoNc) of the two neighbors. For example, since we're given that the left neighbor of $\\frac{3}{7}$ in $F_8$ is $\\frac{2}{5}$, the next fraction to appear between the two will be\n",
"Naturally, this fraction will first appear in $F_{12}$, meaning $\\frac{5}{12}$ is the left neighbor of $\\frac{3}{7}$ in that Farey sequence. We could then, in turn, find the mediant of $\\frac{5}{12}$ and $\\frac{3}{7}$ to find the next left neighbor of $\\frac{3}{7}$ ($\\frac{8}{19}$, appearing in $F_{19}$).\n",
"\n",
"We can repeat this process until we find a mediant with a denominator greater than 1000000. At that point, we know that mediant will *not* be in $F_{1000000}$, so whatever left neighbor we just used to calculate it must be the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
"\n",
"To compute this by hand, observe that the $n$th mediant computed in this manner is simply\n",
"$$\\frac{2 + 3n}{5 + 7n}$$\n",
"Therefore, we just need to find the largest $n$ such that $5 + 7n \\leq 1000000$, which is $n=142856$. This gives us a numerator of 428570.\n",
"* Numerators of Farey sequences: [A007305](https://oeis.org/A007305)\n",
"\n",
"#### Copyright (C) 2025 filifa\n",
"\n",
"This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
