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Extra Egyptians (Posted on 2013-02-14) Difficulty: 3 of 5

(1) Consider Egyptian Number.

(2) Conjecture 1. There is no number greater than 125 that cannot be expressed as a series of fractional terms (1/a+1/b+1/c...=1) so that at least one partial sum of those terms is either 1/2 or 1/3.

(3) Conjecture 2. There is at least one finite set {p1,p2,p3,...} of prime factors of the denominators of the fractional terms so that all integers greater than 125 can be so expressed (it seems that this would have to follow, but can it be proved?)

(4) Can the limit 125 in (2) or (3) be further sharpened?

(5) Is there a further limit beyond which both (2) and (3) can be made to hold simultaneously for all integers? If so, what is it?

No Solution Yet Submitted by broll    
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  Subject Author Date
SolutionMost of a solution; I think this is what is wantedCharlie2013-02-14 17:03:00
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