What non-integer number has the smallest factorial?

Question

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally.

Now, for all positive numbers:

x < y => x! < y!

But not for numbers between 0 and 1. 0! = 1!. I played around to find this:

0!      1
0.1!    0.951
0.4!    0.887
0.425!  0.886
0.45!   0.885
0.5!    0.886
0.55!   0.888
0.6!    0.893
0.9!    0.96
1!      1

What is the floor for the gamma function (or the unrestricted by integers factorial function)? The floor is around 0.45 from the table. I'm curious to know what is the floor value and the number for which gamma is minimum.

Any significance or implications of these numbers would be interesting too.

Compare http://math.stackexchange.com/questions/3245/minimum-of-the-gamma-function-gamma-x-for-x0-how-to-find-x-min — Martin R, Dec 10 '14 at 18:50
That's baffling! Why would 1.5! be less than 1? Besides, calculator says otherwise. — kBisla, Dec 10 '14 at 18:56

score 2 · Accepted Answer · answered Dec 10 '14 at 18:55

As Martin R says, there is no known simple closed form for the minimum of the Gamma function. However, for fun, here are the first 1000 digits of the root (in Mathematica):

FindRoot[D[Gamma[x], x], {x, 1.5}, WorkingPrecision -> 1000]

which produces

1.46163214496836234126265954232572132846819620400644635129598840859878\
6440353801810243074992733725592750556793365533053341617365778466985829\
1771683816450246525426187920443843819783335597739619760747194319349371\
7541405945193010996372416652777217279167325088046396007693297814490147\
5185803414306536810631010706016949785457933765577116113646852653864407\
7372589890682262958196750529119944311972207258664056482074952272808066\
6492780264672546913947612363653574355170333094944302512419288581347767\
7638037268207228647702315131118425758160327915893181546219377794692453\
7619116776460200864228324583987363497206740454783034481431732283690209\
3646770017559017693843888399883958228679946842407928700859042045977138\
8194093679122118848402487784207298777528715900614407383331513574002791\
5353912845037515421736664287138645800903908013264994637024811461027624\
7799875927238664920666176370867887038347260422463147415264091765916362\
0499923428977096412741183720620861677533192913168549317959136215151559\
088748352371795155035

What non-integer number has the smallest factorial?

1 Answers1