Dirac's Nested Square Roots: Making Any Natural Number from Four 2s

The puzzle invites mathematicians to construct natural numbers using exactly four 2s and any mathematical operations. Different mathematical knowledge enables different ranges: elementary school arithmetic handles some examples, middle school techniques like exponents and factorials extend further, and advanced methods involving the Gamma function or complex numbers reach even higher numbers. The number 7 proved notoriously difficult to reach with basic approaches.

Mathematicians in the 1920s treated this as a recreational pursuit until Paul Dirac discovered a universal method for generating any natural number. His solution relies on nested square roots of 2 applied repeatedly, combined with base-2 logarithms, creating an elegant formula. The method natively requires only three instances of the digit 2, not four. However, this constraint yields easily to modification: replacing one digit with a formula that introduces another 2 produces exactly four 2s. With this adjustment, any natural number becomes expressible using four 2s and these mathematical operations, though the actual notation involves many layers of nested square roots.