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.

(In reply to Possible approach by broll)

If one can form a finite list such as broll suggests, for example that all perfect squares are Egyptian and various manipulations to give this finite list, then other combinations come to mind.

Twice the sum of any two Egyptian numbers is Egyptian.
Three times the sum of any three Egyptian numbers is Egyptian.
Four times the sum of any four Egyptian numbers is Egyptian.
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Half the difference between two Egyptian numbers that have been formed from one of the even (twice, four times, etc.) rules above is Egyptian.

One third the difference between any two Egyptian numbers that have been formed from any of the divisible-by-three (three times, six times, ...) is an Egyptian number.

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Is this enough to assure an uninterrupted set beyond 23?

Posted by Charlie on 2013-02-10 09:55:52