Puzzle 4: Solution

Q     Using each of digits 2, 5, 7 once, and only basic arithmetic operations, write a formula for 181.

A     181 = sqrt( sqrt(sqrt(sqrt(25!))) - 7 ).

The main point of this puzzle is the fact that 215-7 is a perfect square. This fact was noted at least as early as 1913, when Ramanujan asked whether 2n-7 can be a square for any integer n>15. That this never happens again was finally proved two generations later by Nagell (Arkiv för Matematik 4 (1960), 185-187).

This curiosity shows up at least twice in research mathematics: in arithmetic algebraic geometry, it lets one show that a certain model of the Klein quartic curve mod 2 is the unique curve of genus 3 with the maximal number of points over the field of 213 elements (see my article on the Klein quartic in “The Eightfold Way”); and in coding theory, it seems to promise a perfect 2-error-correcting binary code of length 91, which unfortunately cannot exist due to a further integrality criterion (see for instance MacWilliams and Sloane's book on error-correcting codes).