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.

Related material Read more...

	Funny Arithmetic
	A single formula to express all numbers
	Representation of numbers with a single 4. The rules and the possibilities
	One 4, a story
	Three 3's
	Representation of numbers with three 4's
	Representation of numbers with three 5's
	Representation of numbers with four 3's
	Representation of numbers with four 4's
	Representation of numbers with four 5's
	A problem of representing 4 by four identical digits
	A problem of representing 6 by three identical digits
	A 9's Fan's Clock
	Make an Identity
	Fun with numbers: place plus/minus signs between the digits
	24 with One Digit
	Representation of 3 with Three Equal Digits

72199682