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One example could be the sphere packing problem in $\mathbb{R}^8$, recently finished by Viazovska. It was a well-known conjecture that $E_8$-lattice packing gives the maximum density for sphere packings in $\mathbb{R}^8$. In 2003, Cohn and Elkies have shown using linear programming that the optimal density is less or equal to 1.000001 times the density of $E_8$ packing. Viazovska's paper finally removed this 1.000001 factor. While I would not call her paper very long, it definitely has some highly non-trivial ideas in it (the appearance of modular forms is pretty surprising to me, for example).