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From: A-Man on 22 Jun 2010 23:24
Hello all... I have a problem which I have been grappling with for some time. Let b be a positive integer and consider the equation z = x + y + b where x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in two classes. I wish to show that at least one of the classes contains a solution to the equation.

I have tried using induction on b. The case b = 1 has been solved entirely by me. But I cannot understand how to use the induction hypothesis to prove the result. The more I think of it, the more I feel that a different approach to the problem is needed, but I cant figure out what. It is sort of a special case of a research problem, which has been solved in a more general way. I have little experience of doing research on my own, and so will be glad if anyone can offer me any advice or hints. Thanks.
From: Gerry Myerson on 23 Jun 2010 19:10
In article
<23089844.12545.1277277907737.JavaMail.root(a)gallium.mathforum.org>,
A-Man <shahabfaruqi(a)gmail.com> wrote:

> Hello all... I have a problem which I have been grappling with for some time.
> Let b be a positive integer and consider the equation z = x + y + b where
> x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in
> two classes. I wish to show that at least one of the classes contains a
> solution to the equation.
>
> I have tried using induction on b. The case b = 1 has been solved entirely by
> me. But I cannot understand how to use the induction hypothesis to prove the
> result. The more I think of it, the more I feel that a different approach to
> the problem is needed, but I cant figure out what. It is sort of a special
> case of a research problem, which has been solved in a more general way. I
> have little experience of doing research on my own, and so will be glad if
> anyone can offer me any advice or hints. Thanks.

Get a copy of Guy, Unsolved Problems In Number Theory, and look at
Problem E14 and all the references therein. There has been a lot of
work on this kind of problem.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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