Sunday, 4 September 2011

soft question - Theorems with unexpected conclusions

I am interested in theorems with unexpected conclusions. I don't mean
an unintuitive result (like the existence of a space-filling curve), but
rather a result whose conclusion seems disconnected from the
hypotheses. My favorite is the following. Let f(n) be the number of
ways to write the nonnegative integer n as a sum of powers of 2, if
no power of 2 can be used more than twice. For instance, f(6)=3
since we can write 6 as 4+2=4+1+1=2+2+1+1. We have
(f(0),f(1),dots)= (1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,dots). The
conclusion is that the numbers f(n)/f(n+1) run through all the
reduced positive rational numbers exactly once each. See A002487 in
the On-Line Encyclopedia of Integer Sequences for more
information. What are other nice examples of "unexpected conclusions"?

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