Making any integer with four 2s

Paul Dirac devised a general method for expressing any positive integer using exactly four instances of the digit 2, resolving a puzzle that had entertained mathematicians since the 1920s. His approach relies on repeated nested square roots combined with base-2 logarithms, since a square root applied n times equals the number raised to the power of 1/2^n, and logarithms then extract the exponent. The construction technically uses only three 2s, but a fourth can be added by replacing a single digit with a representation such as 2/2, keeping the count at exactly four. The method works for any target number, though the resulting expressions grow unwieldy as the number of nested square roots increases. The puzzle and Dirac's solution are recounted in Graham Farmelo's biography The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius.