51st Putnam 1990

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Problem A2

Can we find a subsequence of { n1/3 - m1/3 : n, m = 0, 1, 2, ... } which converges to √2?

 

Solution

Answer: yes.

There is obviously nothing special about √2. In fact a little thought suggests that the set { n1/3 - m1/3 : n, m integers } is dense in the reals.

To prove this note that we can find arbitrarily small values because certainly for n > 64, we have (n1/3 + 1/n)3 < n + 3/4 + 3/1024 + 1/643 < n + 1. Also if α is in the set, then so are all integral multiples of α, because (n3N)1/3 - (n3M)1/3 = n( N1/3 - M1/3). So by taking a suitable multiple of a value less than ε we can get within ε of any desired value.

 


 

51st Putnam 1990

© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001