"Jeremy Boden" <email@example.com> wrote in message news:firstname.lastname@example.org... > 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). >> Your theorem follows, q.e.d. > > 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. KISS-please. :-)
> >> >> 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 > >