An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1/2+1/3+1/9+1/18=1 . Prove that all integers greater than 23 are Egyptian.

Start by looking at small values that can be so expressed:

(1)

Let a=2, b=3
1/1 (1)..1
1/a+1/a...4
1b+1/b+1/b..9
1/a+1/2a+1/2a...10
1/a+1/b+1/ab...11
1/2a+1/2a+1/2a+1/2a...16
1/2a+1/2a+1/b+1/2ab...17
1/b+1/b+1/2b+1/2b...18
1/a+1/2b+1/2b+1/2b..20
1/a+1/2a+1/4a+1/4a..22
1/a+1/2a+1/ab+1/2ab..24
1/5+1/5+1/5+1/5+1/5(5)....25

(2)

If the Egyptian number,n, is a square, we can form it at once using n copies of its reciprocal. If not, then we start with, say, the range 25 to 125 inclusive, and establish that all numbers in that range satisfy the required condition (it's quite possible that a much shorter range would suffice). It's not necessary to actually do the computation, because it is Sloane A125726. Now, using the fact that 1/2+1/4+1/4=1/2+1/3+1/6=1/2+1/5+1/5+1/10=1/3+1/3+1/5+1/15+1/15=1, we can always find the representation of a new number by taking a known solution and partitioning one or more of its segments to achieve the desired total.

(3)

A similar question is whether there is any number greater than 25 that can't be expressed in this way so that at least one partial sum of its terms is either 1/2 or 1/3? There is none between 26 and 55.

 

 

Edited on February 11, 2013, 12:32 am

Posted by broll on 2013-02-10 06:59:36