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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

568.0. "Properties of LSDigits of N!" by MODEL::YARBROUGH () Wed Aug 20 1986 17:42

    After n=2, the least significant nonzero digit of n! is one of
    {2,4,6,8}.
    
    Is the sequence of least significant nonzero digits in n! periodic?
    
    If so, what is its period?
    
    (I don't yet know the answer. There is no periodic behavior up to
    n=52, which is as high as I am usually interested in. I suspect
    it is periodic, with a period length in the range 100-400.)
                                                                   
    Lynn Yarbrough

T.R	Title	User	Personal Name	Date	Lines
568.1		CLT::GILBERT	eager like a child	`Fri Aug 22 1986 21:33`	32
	It's not periodic. In the following, LSD(x) denotes the least-significant non-zero digit of x. Suppose the LSDs of factorials are periodic, with period p, so that LSD((N+p)!) = LSD((N+kp)!), for sufficiently large N. Let q = p2^c, so that LSD(p) is not 5. We have: LSD((q10^b-1)!) = LSD((2q10^b-1)!), and LSD((q10^b)!) = LSD((2q10^b)!) But LSD(( q10^b)!) = (LSD( q ) LSD(( q10^b-1)!)) mod 10, and LSD((2q10^b)!) = (LSD(2q) * LSD((2q10^b-1)!)) mod 10, so that LSD(q) * x = LSD(2q) x, modulo 10, where x = 2,4,6 or 8. However, this implies that LSD(q) = LSD(2q), modulo 5. (if for no other reason than the following modulo 10 multiplication table: 1 2 3 4 6 7 8 9 ----------------------- 2 4 6 8 2 4 6 8 4 8 2 6 4 8 2 6 6 2 8 4 6 2 8 4 8 6 4 2 8 6 4 2 ). But LSD(q) = LSD(2q) modulo 5 gives a contradiction, since we chose q so that LSD(q) was not 5: LSD( q ) 1 2 3 4 6 7 8 9 LSD(2*q) 2 4 6 8 2 4 6 8 Thus, the assumption of periodicity must be false.
568.2	map LSD's onto a square-free string in {1,2,3}?	REGINA::OSMAN	and silos to fill before I feep, and silos to fill before I feep	`Wed Sep 03 1986 18:34`	6
	Gee, if it's not periodic, perhaps there's a simple mapping of the LSD's onto {1,2,3} to give us a sequential method of generating a square free string in {1,2,3}. /Eric
568.3	Here's LSDs for 1! through 100000!	REGINA::OSMAN	and silos to fill before I feep, and silos to fill before I feep	`Mon Sep 08 1986 17:27`	23
	If the LSD of n! is not periodic, I find it intriguing to see if there is any obvious way to generating the pattern. Towards this end, here are the LSD's of 1! through 100000!, using Gilbert's compaction routine: 1264224288868238682448465318266264448464484672428448463868222428224286626438 6(28,22)886821626444846948462242816(2,13)886829484699(52,23)72428448461626466 26488682(50,25)94846886827(61,14)1(1,25)(76,24)(50,75)(75,25)22428448466 (186,14)(25,25)6(126,14)(265,60)(125,25)33(102,23)(450,15)8868294846(450,26) (26,24)(400,26)(76,24)(600,50)(450,75)(75,50)(175,25)(100,25)4(226,24) (100,25)8(76,49)2(176,24)(250,50)(100,25)(225,25)(450,50)(600,50)(250,125) (125,25)(50,25)(175,25)(150,25)(1125,51)(26,99)(375,51)(676,74)(500,50) (100,25)(575,25)(250,50)(400,100)(1500,125)(175,25)(550,25)(925,125)(600,50) (1100,76)(426,74)(400,50)(50,1200)(550,25)(2275,100)(1375,1125)(500,250) (650,50)(300,950)55(3752,1248)(1500,125)(125,1125)77(5252,123)(1375,1125) (1000,125)(125,1125)(150,25)(1775,225)(2500,125)(1625,875)(250,26)(26,12474) (2500,1250)(1250,11250)(3750,50)(10050,1200)(1250,11250)(5000,1250) (1250,11250)5513(37504,12496)(7500,1250)(1250,11250)(8750,26)(26,12474) Can anyone find a way to describe the pattern ? (I can't yet. This isn't a quiz) /Eric