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How to prove or disprove this statement:

For all $c, there exists $0, $0\leq j, such that all conditions hold simultaneously:

  1. $z=is+i(i-1)/2-j-k(s+i+2j)-k(k-1)/2$,
  2. $z and $0
  3. for all $0\leq m and $0, a) holds

a) $ms+m(m-1)/2\neq is+i(i-1)/2-j-n(s+i+2j)-n(n-1)/2$

All variables are integers. I have tried computationally to find a counterexample but without luck.


This would imply the existence of several Self-avoiding walk on $\mathbb{Z}$ . (but not the converse.)

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    Since the "Self-avoiding walk" problem (the motivation for the OP) has been solved on mathoverflow without any ugly equations, I think we may safely declare this question completely unintereseting.2012-03-04

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