You are told that all integers from 33 to 73 inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than 73.
Solution
The trick is consider the integers 2a1, 2a2, ... , 2am given that a1, a2, ... , am is a solution for n. The sum of their reciprocals is 1/2. So if we throw in two 4s, we get a solution for 2n + 8. Similarly, adjoining 3 and 6 gives a solution for 2n + 9. It is now a simple induction. For the starter set gives 74 thru 155, then those give 156 thru 319, and so on. In general, n thru 2n+7 gives 2n+8 thru 4n+23 = 2(2n+8) + 7.
© John Scholes
jscholes@kalva.demon.co.uk
11 May 2002