Do there exist 1985 distinct positive integers such that the sum of their squares is a cube and the sum of their cubes is a square?
Solution
Answer: yes.
Take any n integers ai. Suppose that ∑ ai2 = b and ∑ ai3 = c. Now multiply each ai by b4c3. The sum of their squares becomes b9c6 which is a cube and the sum of their cubes becomes b12c10 which is a square.
© John Scholes
jscholes@kalva.demon.co.uk
26 Aug 2002