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Solutions of a particular equation 
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#1
Jun2310, 02:24 AM

P: 8

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. 


#2
Jun2510, 10:56 AM

Sci Advisor
P: 1,742

A class C of {1,2,...,4b+5} contains a solution is equivalent to that x,y,z are elements of C ?



#3
Jun2510, 08:42 PM

P: 894




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