Re: Re: All Factors of a number

*To*: mathgroup at smc.vnet.net*Subject*: [mg53094] Re: [mg53059] Re: All Factors of a number*From*: János <janos.lobb at yale.edu>*Date*: Thu, 23 Dec 2004 07:58:41 -0500 (EST)*References*: <200412201134.GAA02639@smc.vnet.net> <200412211019.FAA17308@smc.vnet.net> <200412220952.EAA04441@smc.vnet.net> <opsje7w8pwiz9bcq@monster.ma.dl.cox.net>*Sender*: owner-wri-mathgroup at wolfram.com

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 >

**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>

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**Re: Re: All Factors of a number**

**Re: Re: All Factors of a number**

**Re: Re: All Factors of a number**