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Re: Re: All Factors of a number
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
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