table of differences between $\\delimiter"4264306\\sqrt{n!}\\delimiter"5265307^{2}$ and $n!$ for $0<n<26$

There are only three known solutions to Brocard’s problem, and the near misses all seem to occur early on. Notice how, for example, 3! is just 3 shy of a square (compared to 1 shy of a square which is what Brocard’s problem asks for). Still, the differences between a factorial and the next higher perfect square don’t make for a consistently ascending order sequence. For a few values of $n$ , (such as 4, 7, 10, 24, 26, 42, 117, 135) this difference is smaller than the previous difference. In general, however, the difference between a factorial and the next perfect square widens as $n$ gets larger.

The following table gives the square root of $n!$ to six decimal places, and then the difference between the factorial and the next higher square (obtained by taking the ceiling of the square root of $n!$ and squaring that integer).

单词	TableOfDifferencesBetweenlceilsqrtnrceil2AndNFor0N26
释义	table of differences between $\\delimiter"4264306\\sqrt{n!}\\delimiter"5265307^{2}$ and $n!$ for $0<n<26$ There are only three known solutions to Brocard’s problem, and the near misses all seem to occur early on. Notice how, for example, 3! is just 3 shy of a square (compared to 1 shy of a square which is what Brocard’s problem asks for). Still, the differences between a factorial and the next higher perfect square don’t make for a consistently ascending order sequence. For a few values of $n$ , (such as 4, 7, 10, 24, 26, 42, 117, 135) this difference is smaller than the previous difference. In general, however, the difference between a factorial and the next perfect square widens as $n$ gets larger. The following table gives the square root of $n!$ to six decimal places, and then the difference between the factorial and the next higher square (obtained by taking the ceiling of the square root of $n!$ and squaring that integer).
随便看	Poincaré duality Poincaré formula Poincaré lemma Poincaré recurrence theorem Poincaré’s theory of Fuchsian groups Point point PointAndACompactSetInAHausdorffSpaceHaveDisjointOpenNeighborhoods point and a compact set in a Hausdorff space have disjoint open neighborhoods. PointCountableBase point countable base PointedSet pointed set PointedTopologicalSpace pointed topological space Pointfinite point-finite PointfreeGeometry point-free geometry PointPreventingUniformConvergence point preventing uniform convergence Pointwise pointwise PointwiseConvergence pointwise convergence

table of differences between \\delimiter⁢"⁢4264306⁢n!⁢\\delimiter⁢"⁢52653072 and n! for 0<n<26

table of differences between $\\delimiter"4264306\\sqrt{n!}\\delimiter"5265307^{2}$ and $n!$ for $0<n<26$