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.

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

The solution relies on a basic property of logarithms: log_ab^c = c·log_ab:

N = - log₂log₂√√...√2,

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

1 = - log₂log₂√2,
2 = - log₂log₂√√2,
3 = - log₂log₂√√√2,
4 = - log₂log₂√√√√2,
and so on.

Indeed,

√√...√2 = 2^{2^-N,}

wherefrom,

log₂√√...√2, = 2^-N.

And, finally,

log₂2^-N = -N.

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