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	integrity characterized by places InteractiveAnimationGraphics interactive animation graphics InterceptTheorem intercept theorem Intercession intercession Interest interest InterestRate interest rate Interior interior InteriorAxioms interior axioms InteriorPointOfSetInTopologicalSpace interior point of set in topological space InterleaveSequence interleave sequence IntermediateValueTheorem intermediate value theorem IntermediateValueTheoremForExtendedRealNumbers intermediate value theorem for extended real numbers InternalDirectSumOfIdeals internal direct sum of ideals

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 $0<n<26$