Able to solve this simple-looking math problem?
dan.ms...@gmail.com
simply that k is also a natural number ? Seems to make sense , but
I've been unable to prove it , or find a counterexample .
Dave L. Renfro
The same idea behind a well-known proof that log(2) (base 10)
is irrational (google it) can be used to show that if 2^k is
not an integer power of 2, then k is irrational (hence k is
certainly not a natural number).
Dave L. Renfro
The World Wide Wade
Robert Israel
> dan.ms.ch...@gmail.com wrote:
>
> > 2^k is a natural number =A0and 3^k =A0is a natural number =A0.
> > Does this simply that k is also a =A0natural number ? Seems
> > to make sense , but I've been unable to prove it , or
> > find a counterexample .
>
> The same idea behind a well-known proof that log(2) (base 10)
> is irrational (google it) can be used to show that if 2^k is
> not an integer power of 2, then k is irrational (hence k is
> certainly not a natural number).
Ummm... "2^k is not an integer power of 2" means k is not a natural
number. That's not the question. The question here is whether there
are natural numbers x and y such that log_2(x) = log_3(y) and is
irrational. IIRC it's a very hard problem.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Dave L. Renfro
>> The same idea behind a well-known proof that log(2) (base 10)
>> is irrational (google it) can be used to show that if 2^k is
>> not an integer power of 2, then k is irrational (hence k is
>> certainly not a natural number).
The World Wide Wade wrote:
> What about 2^(3/2)?
Ooops!
Dave L. Renfro
Dave L. Renfro
> Ummm... "2^k is not an integer power of 2" means k is not a natural
> number. That's not the question. The question here is whether there
> are natural numbers x and y such that log_2(x) = log_3(y) and is
> irrational. IIRC it's a very hard problem.
Yep, I misread the question.
Dave L. Renfro
amy666
> <eef632fb-b065-4520...@m3g2000hsc.googl
sqrt(8) ?
Gerry Myerson
<8e7818de-c99a-4660...@l42g2000hsc.googlegroups.com>,
dan.ms...@gmail.com wrote:
You asked the same question in this newsgroup in March.
What don't you understand about the answers posted then?
Back then, the subject header was,
Can you solve this math problem? Please help .
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Angus Rodgers
<ge...@maths.mq.edi.ai.i2u4email> wrote:
>In article
><8e7818de-c99a-4660...@l42g2000hsc.googlegroups.com>,
> dan.ms...@gmail.com wrote:
>
>> 2^k is a natural number and 3^k is a natural number . Does this
>> simply that k is also a natural number ? Seems to make sense , but
>> I've been unable to prove it , or find a counterexample .
>
>You asked the same question in this newsgroup in March.
>What don't you understand about the answers posted then?
>
>Back then, the subject header was,
>Can you solve this math problem? Please help .
I missed that thread in March. The mention of the 1971 Putnam problem
rang a faint bell, which turned out to be this ancient thread from 1993:
<http://groups.google.co.uk/group/sci.math/browse_frm/thread/fb1ca767d5f21e45/>
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
Han de Bruijn
Is the problem still open ?
Han de Bruijn
dan.ms...@gmail.com
> What don't you understand about the answers posted then?
...
wrote:
>Although I can't seem to find that post again .
<http://groups.google.com/group/sci.math/msg/35bb7db3b2b6d36a?hl=en>
Angus Rodgers
dan.ms...@gmail.com wrote:
>Although I can't seem to find that post again .
<http://groups.google.co.uk/group/sci.math/browse_frm/thread/6360d3b952db829b/>
Gerry Myerson
Han de Bruijn
I'm interested because it (vaguely) resembles a strange finding of mine
which is found at:
http://hdebruijn.soo.dto.tudelft.nl/www/grondig/calculus.htm
There are two variations of idealized Multigrid Calculus, namely Double
Grid Calculus, and Triple Grid Calculus. With the first method, you can
do grid coarsening and grid refinement with factors 2^k where k integer
and with the second method you can do grid coarsening & grid refinement
with factors 3^k where k integer. You can combine the two, but there are
_no_ other possibilities, to the best of my knowledge:
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf : Double Grid
http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf : Triple Grid
Han de Bruijn
amy666
> In article
> <8e7818de-c99a-4660...@l42g2000hsc.goog
> legroups.com>,
> dan.ms...@gmail.com wrote:
>
> > 2^k is a natural number and 3^k is a natural
> number . Does this
> > simply that k is also a natural number ? Seems to
> make sense , but
> > I've been unable to prove it , or find a
> counterexample .
>
> You asked the same question in this newsgroup in
> March.
> What don't you understand about the answers posted
> then?
answers ? the problem is not solved !
no solution was posted and the question is in fact still open today.
the matter is far from settled.
its a very well-known open problem , however i dont know what it is actually called ...
is there a name or an inventor or is it to old and simple to receive a name or credit an inventor ?
it relates to the four-exponential conjecture but its not equivalent ... what is this called then ?
im not even sure about the exact category this problem lies in ...
is this number theory or algebra ?
i once had the following idea very very long ago
consider the ring T :
a + b sqrt(2) + c sqrt(3) + d sqrt(6)
where a b c and d are non-negative rationals.
let A B C D be elements of T
where A and B are given fixed elements.
if A ^ k = C and B ^ k = D
does this imply k is an element of T too ?
more specific : does this imply k is rational ??
***
i considered the question trivial and logical and at first assumed i could solve it or at least reduce it to a simple irrationality question like is log(2) irrational.
since k cannot be algebraic deg >= 2 ( e.g. 1 + sqrt(2) ^ sqrt(3) does not reduce ) i of course quickly restated to " does this imply k is rational ".
i considered reducing logaritms e.g. log(1+sqrt(2))
to a form x + y sqrt(2) in analogue to taking the complex logaritm of e.g. log ( 4 + 3 i) = x + y i
the analogue functions of arctan ( appearing in complex logaritms ) where named " splitlogs ".
and using taylor series and others , they were computable.
however they did nothing at resolving the question.
they only reminded me of multisections.
i also considered extending the ring and then again try to apply the "splitlogs " to resolve the matter.
it also failed.
then i started to consider the absolute values.
the absolute values had to match so in a way , a neccessary but insufficient condition.
i thus assumed i was getting closer.
but even for just the positive integers ( abs ) instead of the ring T , i failed to prove.
that problem is as good as equal to the OP's simple looking math problem.
although i assumed at first i was getting closer , i was wondering if for non-rational elements of T this was really true ??
maybe i was getting further away instead ???
i spend a lot of time on this matter and considered it " understanding ring theory better " and " as important as RH ! "
at that time i wrote 3 big question marks.
the first for the conjecture.
the second was the assumption that most mathematicians , especially ring theorists and number theorists could easily solve such problem , so why couldnt i even find a clue and why cant i find this " simple proof " on the internet or in a book ? not even a clue ??
how could i have good grades in math and not be able to solve this " simple problem " ?
and the third question mark :
because of my desire to find the solution in a book or to solve it myself , but with a hint from a book, i was wondering to which category the problem belonged.
today i know it relates to the four exponentials conjecture , but im still not sure exactly what category this falls into or what this problem is named.
because of the way i arrived at the conjecture , the ring T ;
i labeled it " ring theory ? " ( the 3rd question mark )
i hoped to learn about a solution during education.
dream on tommyboy !
the " logsplits " are still being considered for various (algebraic) rings.
( even for rings not proven to exist see my posts about " transcendental rings " )
i like the aesthetics ...
math can be a belief, an art and an obsession ... rather then science, knowledge and logic ...
>
> Back then, the subject header was,
> Can you solve this math problem? Please help .
>
> --
> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for
> email)
regards
tommy1729
Gerry Myerson
That's as may be, but the number theory problem really isn't about
2^k and 3^k. You could just as well ask whether 42^k and 65^k
both being natural numbers implies that k is a natural number. It's
probably no harder, and no easier, to answer this question, than
to answer the question about 2^k and 3^k.
Gerry Myerson
<29897646.1216328972...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
> Gerry Myerson wrote :
>
> > In article
> > <8e7818de-c99a-4660...@l42g2000hsc.goog
> > legroups.com>,
> > dan.ms...@gmail.com wrote:
> >
> > > 2^k is a natural number and 3^k is a natural number . Does this
> > > simply that k is also a natural number ? Seems to make sense , but
> > > I've been unable to prove it , or find a counterexample .
> >
> > You asked the same question in this newsgroup in March. What don't
> > you understand about the answers posted then?
>
> answers ? the problem is not solved !
Noting that the problem is not solved is an answer.
> no solution was posted and the question is in fact still open today.
>
> the matter is far from settled.
>
> its a very well-known open problem , however i dont know what it is actually
> called ...
>
> is there a name or an inventor or is it to old and simple to receive a name
> or credit an inventor ?
>
> it relates to the four-exponential conjecture but its not equivalent ... what
> is this called then ?
It is a consequence of the four exponentials conjecture.
It is one of (infinitely) many consequences of the four exponentials
conjecture. I wouldn't expect each of them to have a name.
> im not even sure about the exact category this problem lies in ...
>
> is this number theory or algebra ?
I'd say it's diophantine analysis, which is a part of number theory.
But categorical distinctions are of limited usefulness. For all we know,
the solution will pop out of someone's work on partial differential
equations, and then where will we be?
> i once had the following idea very very long ago
>
> consider the ring T :
> a + b sqrt(2) + c sqrt(3) + d sqrt(6)
>
> where a b c and d are non-negative rationals.
That's not a ring - it doesn't have additive inverses.
> let A B C D be elements of T
>
> where A and B are given fixed elements.
>
> if A ^ k = C and B ^ k = D
>
> does this imply k is an element of T too ?
Of course not, e.g., A = B = 2, C = D = 3.
Have you ever looked at Schanuel's Conjecture?
I think it encompasses everything you want to do in T,
and more, and correctly.
Gerry Myerson
Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
> In article
> <29897646.1216328972...@nitrogen.mathforum.org>,
> amy666 <tomm...@hotmail.com> wrote:
>
> > Gerry Myerson wrote :
> >
> > > In article
> > > <8e7818de-c99a-4660...@l42g2000hsc.goog
> > > legroups.com>,
> > > dan.ms...@gmail.com wrote:
> > >
> > > > 2^k is a natural number and 3^k is a natural number . Does this
> > > > simply that k is also a natural number ? Seems to make sense , but
> > > > I've been unable to prove it , or find a counterexample .
> > >
> >
> > the matter is far from settled.
> >
> > its a very well-known open problem , however i dont know what it is
> > actually called ...
> >
> > is there a name or an inventor or is it to old and simple to receive a name
> > or credit an inventor ?
I have found that in 1944 Alaoglu and Erdos asked, If p and q are
both primes, is it true that p^x and q^x are both rational only if x
is an integer?
I don't know whether they were the first to raise this question.
amy666
> In article
> <29897646.1216328972...@nitrogen.math
makes sense , but still this specific example is so simple , i think it ought to have a name :)
>
> > im not even sure about the exact category this
> problem lies in ...
> >
> > is this number theory or algebra ?
>
> I'd say it's diophantine analysis, which is a part of
> number theory.
> But categorical distinctions are of limited
> usefulness. For all we know,
> the solution will pop out of someone's work on
> partial differential
> equations, and then where will we be?
i agree.
>
> > i once had the following idea very very long ago
> >
> > consider the ring T :
> > a + b sqrt(2) + c sqrt(3) + d sqrt(6)
> >
> > where a b c and d are non-negative rationals.
>
> That's not a ring - it doesn't have additive
> inverses.
pardon me , a semi-ring of course.
>
> > let A B C D be elements of T
> >
> > where A and B are given fixed elements.
> >
> > if A ^ k = C and B ^ k = D
> >
> > does this imply k is an element of T too ?
>
> Of course not, e.g., A = B = 2, C = D = 3.
yes evidently but of course i meant A B C D distinct elements of T.
>
> Have you ever looked at Schanuel's Conjecture?
> I think it encompasses everything you want to do in
> T,
> and more, and correctly.
yes i did look at it , but it is not proven.
im not sure it encompasses everything i want to do in T.
as for your " and correctly " i dont think i made any big mistakes in my posts , apart from saying ring instead of semi-ring , did i ?
>
> --
> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for
> email)
regards
tommy1729
Gerry Myerson
<11451812.1216409604...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
Well, there was that, and there was not saying you meant
A, B, C, and D to be distinct, and then there's A=2, B=4, C=3, D=9,
but, you know, it's not my responsibility to find mistakes in your work,
that's what you should do, preferably before you post.