# Any Integer with Three 2s

A problem at the 1936 Moscow Mathematical Olympiad had been credited to Paul Dirac:

A single formula to express every positive integer with three 2s and any kind of mathematical symbols.

You may want to try to wrok out the problem yourself before checking the solution.

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Copyright © 1996-2018 Alexander BogomolnyA single formula to express every positive integer with three 2s and any kind of mathematical symbols.

### Solution

The solution relies on a basic property of logarithms: log_{a}b^{c} = c·log_{a}b:

N = - log_{2}log_{2}√√...√2,

where the number of radicals is exactly N. For example,

1 = - log_{2}log_{2}√2,

2 = - log_{2}log_{2}√√2,

3 = - log_{2}log_{2}√√√2,

4 = - log_{2}log_{2}√√√√2,

and so on.

Indeed,

√√...√2 = 2^{2-N,
}

wherefrom,

log_{2}√√...√2, = 2^{-N}.

And, finally,

log_{2}2^{-N} = -N.

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Copyright © 1996-2018 Alexander Bogomolny70760585