AI 中文总结
该研究沿用奇波里尼的框架，通过解析证明结合计算机验证，得出了埃尔德什问题1005中f(n)的精确公式，确定了对应常数c的值。
AI 中文摘要
1943年，埃尔德什研究了n阶法里序列中，两个分子分母反向排列的分数之间的项的最小数量f(n)。确定f(n)=(c+o(1))n中的常数c被称为埃尔德什问题1005。近期，奇波里尼（Cipollini）通过证明f(n)=(1/4+o(1))n解决了这个渐近问题。我们沿用其框架，对所有足够大的n给出了f(n)的精确公式的解析证明。结合有限计算机验证，我们进一步确定了每个整数n≥4对应的f(n)。
英文摘要
In 1943, Erdős considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erdős Problem 1005. Recently, Cipollini solved this asymptotic problem by proving that $f(n)=(1/4+o(1))n$. Following his framework, we give an analytic proof of an exact formula for $f(n)$ for all sufficiently large $n$. Combining this with a finite computer verification, we further determine $f(n)$ for every integer $n\geq 4$.
