Andrew Booker and Andrew Sutherland have found, using the the Charity Engine distributed computing platform, a third solution in integers to the equation , so we now know each of

1^{3} + 1^{3} + 1^{3}

4^{3} + 4^{3} + (5)^{3}

569936821221962380720^{3} + (569936821113563493509)^{3} + (472715493453327032)^{3} (verify!)
is equal to 3. It is conjectured that there are infinitely many integer solutions, but we seem to be nowhere near that.