Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Easy problem (Opposite of Goldbach???)
Replies: 23   Last Post: Aug 24, 2006 6:56 PM

 Messages: [ Previous | Next ]
 Tapio Hurme Posts: 853 Registered: 12/8/04
Re: Easy problem (Opposite of Goldbach???)
Posted: Aug 23, 2006 2:55 PM

"Jeremy Boden" <jeremy@jboden.demon.co.uk> wrote in message
news:1156334257.6187.17.camel@localhost.localdomain...
> On Tue, 2006-08-22 at 22:52 -0400, T.H. Ray wrote:
>> > Prove that (for n >= 12) n can be written as the sum
>> > of two different
>> > *non-primes*.

> ...
>>
>> Non-prime integers are called composites. Composites
>> decompose to prime factors. The prime decomposition
>> of 12, e.g., is 2*2*3.
>>
>> The proof that every composite >= 12 can be expressed
>> as the sum of two composites, e.g. 12 = 6+6 (2*3+2*3)
>> relies on the fact that every composite >= 4 decomposes
>> into primes (follows from the fundamental theorem of
>> arithmetic, FTA).
>>
>> Because every composite integer can be expressed
>> in only one way as the product of one or more primes
>> (FTA), it follows by induction that any composite
>> integer can be expressed as the product of two or more
>> composites (because multiplication is repeated addition).

>
> But I didn't ask for the sum of two or more composites (which would be
> trivial).
> I asked for the sum of exactly two composites (which is not quite so
> trivial). The proof does not need FTA, or even the existence of prime
> numbers.
>
> It is always important to read the question first.

Hmmm... T.H.Ray´s methodology is certainly OK, but I - for example - did not
think so formally complicated. I proposed my solution without FTA just
checking the boundary conditions in the problem setting and then the
solution was evident - actually quite easy. I prefer intuition and layman
thinking instead formal methodology. Jeremy writes below " If I wanted easy
problems to appear very difficult then I would do exactly as you suggested".
Iff the problem should have been written so that it appears very difficult,
I hardly could not solve it.

Tapio

>
>>
>> The theorem may be trivial, but the method by which one
>> arrives at the proof is important. This strategy uses
>> induction, and assumes that B|A & C|B, then C|A.
>> That is, given the axiom (Dedekind-Peano) of induction(A)
>> and given that FTA (B) is supported by (A), your theorem
>> (C), q.e.d.
>>
>> That's the heavy lifting. To make the proof more
>> comprehensible and logically coherent, it should be
>> in formal order: definitions-theorem-proof. Try it.
>>
>> If you teach your son to approach a mathematics problem
>> strategically in this way (the method originated with
>> Euclid, from whose work the FTA is also derived), you
>> will have given him a valuable head start on his peers,
>> toward comprehending how mathematics research actually
>> works. For your tutoring effort (which I applaud), I
>> recommend the very accessible book, How to Read and Do
>> Proofs, by Daniel Solow. It is not beyond a bright HS
>> student.
>>

> If I wanted easy problems to appear very difficult then I would do
> exactly as you suggested.
>
> --
> Jeremy Boden
>
>

Date Subject Author
8/18/06 Jeremy Boden
8/18/06 AB
8/18/06 Jeremy Boden
8/18/06 AB
8/18/06 Jeremy Boden
8/18/06 mareg@mimosa.csv.warwick.ac.uk
8/18/06 Dik T. Winter
8/19/06 Phil Carmody
8/19/06 Jeremy Boden
8/19/06 Phil Carmody
8/18/06 Tapio Hurme
8/18/06 Jeremy Boden
8/19/06 Butch Malahide
8/19/06 Proginoskes
8/19/06 Virgil
8/22/06 T.H. Ray
8/23/06 Proginoskes
8/23/06 T.H. Ray
8/23/06 fernando revilla
8/23/06 Jeremy Boden
8/23/06 Tapio Hurme
8/23/06 T.H. Ray
8/24/06 Jeremy Boden
8/24/06 T.H. Ray