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Topic: Reciprocals of integers summing to 1
Replies: 21   Last Post: Nov 23, 2012 8:44 PM

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Luis A. Rodriguez

Posts: 732
Registered: 12/13/04
Re: Reciprocals of integers summing to 1
Posted: Nov 17, 2012 11:52 AM
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El viernes, 16 de noviembre de 2012 00:39:24 UTC-4:30, Charlie-Boo escribió:
> For each n, what are the solutions in positive integers (or in
> integers) to (1/X1)+(1/X2) + . . . + (1/Xn)=1 ?


There are infinitely many different solutions. But I am not sure that it is valid for all n.
Take k primes and do the following sum: 1/p1 + 1/p2 + 1/p3 + ...+ 1/pk = S < 1
Now make Q = 1 - S.
By the theorem of the Egyptian fractions, Q always can be decomposed as:
Q = 1/x1 + 1/x2 + 1/x3 +....+ 1/xj.
I am not sure that it is possible, ever, that k+j = n.
Ludovicus



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