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Prove or disprove:

For all functions $f:\Bbb Z^+ \to \{0,1\}$ , the following is true: There exists positive integers n and d such that: $$f(n)=f(n+d)=f(n+2d)=f(n+3d)$$

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    Is it homework ?2012-05-04
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    No, it is not. I can show that their exists n and d such that f(n)=f(n+d)=f(n+2d). I wanted to know if the statement that I posted originally is true or not.2012-05-04
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    I would say yes, but so far, I have no proof. I think its even true for arbitrary length. (T. Tao would know of course ;))2012-05-04
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    I think it is true too. Moreover I think that that for all positive integers k there exists n and d such that f(n)=f(n+d)=f(n+2d)=f(n+3d)=......=f(n+kd)2012-05-04

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