June 3–June 9, 2008

Problem

Observe that

1 = 1^2, 2 = -1^2 - 2^2 - 3^2 + 4^2, 3 = -1^2 + 2^2, 4 = -1^2 -2^2 + 3^2, 5 = 1^2 + 2^2, 6 = 1^2 - 2^2 + 3^2

This suggests the conjecture: any positive integer n can be expressed in the form

n = e_1 1^2 + e_2 2^2 + ... + e_m m^2

with m a positive integer, and ei = 1 or -1, for i = 1, 2, ..., m. Prove this conjecture.

Solution

here. Peiyush Jain found a way to express n with many fewer terms, though the proof is a bit more involved, here.

List of solvers

Steve Wilmarth (undergraduate); Sean Holman, Doug Faust, Dustin Moody (graduate); Konrad Schroder (staff); Peiyush Jain, Roman Holstein, Kate Smith, Lloyd Sakazaki (outside).

Sean Holman wins the prize!