Re: Re: All Factors of a number

*To*: mathgroup at smc.vnet.net*Subject*: [mg53100] Re: [mg53059] Re: All Factors of a number*From*: DrBob <drbob at bigfoot.com>*Date*: Thu, 23 Dec 2004 07:59:10 -0500 (EST)*References*: <200412201134.GAA02639@smc.vnet.net> <200412211019.FAA17308@smc.vnet.net> <200412220952.EAA04441@smc.vnet.net> <opsje7w8pwiz9bcq@monster.ma.dl.cox.net> <782161BD-E7A5-4763-A46F-1933DDA6E0F8@yale.edu>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

I doubt there are any other doubles or triples, but we need to ask a number theorist. Bobby On Wed, 22 Dec 2004 13:36:08 -0500, János <janos.lobb at yale.edu> wrote: > > > By "double trunk" I meant to have to consecutive rows where the 2^p-1 > has no factors. Like n=6 and 7, or n=7 and 8. To reformat it as a > better mathematical question: Is there two consecutive primes Prime[n] > and Prime[n+1] such that 2^Prime[n]-1 and 2^Prime[n+1]-1 are also > primes for n > 8, that is they have no factors other than themselves > and 1. > > If there would be such numbers, they would give a nice trunk to the > "christmas tree" to stand upright. > > Thanks ahead, > János > > On Dec 22, 2004, at 12:52 PM, DrBob wrote: > > > >> I'm not sure what you mean by a "double trunk", but there are >> Mersennes with two factors at n=5, 9, 12, 13, 19, 23... and others. >> Either I'm misunderstanding you, or numbers are running together on >> your screen. >> >> Timing on my wife's 2GHz dual-G5 is 818.86 seconds... not a huge >> improvement over your G4. It's only 42% faster despite a 60% faster >> clock speed! >> >> Bobby >> >> On Wed, 22 Dec 2004 04:52:49 -0500 (EST), János <janos.lobb at yale.edu> >> wrote: >> >> >>> Bobby, >>> >>> Nice !! >>> >>> You must have a minimum 17 inch monitor as I see :) >>> >>> With the best, >>> >>> János >>> P.S: G4 1.25G Total Time: 1161.98 Second. Not bad for a Mac ! I >>> expected 1277.44 Second. Here is a naive question for you. Will you >>> ever have a "trunk" of length 2, so this tree can stand on it ? If I >>> am >>> counting from the bottom up, the first single trunk is at n=31, there >>> is no double trunk, the only triple is at n={6,7,8} and well the top >>> with length 4 is the top. I skipped lectures when number theory was >>> taught, - which might says, there will be never a double "trunk", may >>> be not even a single one with n > 54 -, so I am not a good candidate >>> to >>> answer it :) >>> >>> On Dec 21, 2004, at 5:19 AM, DrBob wrote: >>> >>> >>> >>> >>>> FactorInteger does this already. >>>> >>>> Just for grins, here's code to list factorizations of Mersenne >>>> numbers >>>> in Xmas tree format. >>>> >>>> toPowers = {{a_Integer, 1} -> HoldForm[a], {a_Integer, b_Integer} -> >>>> HoldForm[a]^HoldForm[b], >>>> List -> Times}; >>>> toStars = StringReplace[ToString[#1 /. toPowers], " " -> " * "] & ; >>>> n = 1; >>>> {totalTime, results} = >>>> Timing[First[Last[Reap[While[n <= 54, p = Prime[n]; >>>> Sow[Timing[{n, p, FactorInteger[2^p - 1]}]]; n++]]]]] /. >>>> {(s_)*Second, {n_, p_, f_}} :> {n, p, s, toStars[f]}; >>>> TableForm[results, TableAlignments -> {Center, Center, Right}, >>>> TableHeadings -> {None, {"n", "p = Prime[n]"*1, Seconds, "Factors of >>>> 2^p-1"}}] >>>> Print["Total Time: ", totalTime] >>>> >>>> That took 499 seconds on my AMD 3200+ machine, so reduce 54 to a >>>> smaller number if you're using a slower machine or don't have the >>>> patience. Of those 499 seconds, 125 were used for the 54th Mersenne, >>>> 201 for the 47th Mersenne, and 63 for the 50th. The rest go pretty >>>> fast, so reducing 54 to 46 will work well. >>>> >>>> Bobby >>>> >>>> On Mon, 20 Dec 2004 06:34:49 -0500 (EST), Ulrich Sondermann >>>> <usondermann at earthlink.net> wrote: >>>> >>>> >>>> >>>>> Following is a test code that I am trying to get down to a set of >>>>> instructions that will allways put all of the factors of a number in >>>>> a table >>>>> or list. >>>>> bt contains all of the factors if I multiply each list entry, >>>>> however >>>>> I >>>>> cannot accomplish that with a single line, as an example I have >>>>> broken into >>>>> the three lists needed for this example. The results of each >>>>> "Times@@" is >>>>> what I am after all placed into one table. All of my attempts have >>>>> proved >>>>> disasterous, I am new to Mathematica and could do this with nested >>>>> loops in >>>>> any programming language, but this has me stumped. >>>>> Thanx! >>>>> >>>>> <<DiscreteMath`Combinatorica` >>>>> bt=Table[KSubsets[{1,2,3,5},a],{a,3}] >>>>> {{{1},{2},{3},{5}},{{1,2},{1,3},{1,5},{2,3},{2,5},{3,5}},{{1,2,3}, >>>>> {1,2,5},{1 >>>>> ,3,5},{2,3,5}}} >>>>> bt1=bt[[1,All]] >>>>> {{1},{2},{3},{5}} >>>>> Table[Times@@bt1[[a]],{a,4}] >>>>> {1,2,3,5} >>>>> bt2=bt[[2,All]] >>>>> {{1,2},{1,3},{1,5},{2,3},{2,5},{3,5}} >>>>> Table[Times@@bt2[[a]],{a,6}] >>>>> {2,3,5,6,10,15} >>>>> bt3=bt[[3,All]] >>>>> {{1,2,3},{1,2,5},{1,3,5},{2,3,5}} >>>>> Table[Times@@bt3[[a]],{a,4}] >>>>> {6,10,15,30} >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>> -- DrBob at bigfoot.com >>>> www.eclecticdreams.net >>>> >>>> >>> >> -- DrBob at bigfoot.com >> www.eclecticdreams.net >> > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**All Factors of a number***From:*"Ulrich Sondermann" <usondermann@earthlink.net>

**Re: All Factors of a number***From:*DrBob <drbob@bigfoot.com>

**Re: All Factors of a number***From:*János <janos.lobb@yale.edu>