
Re: Reciprocals of integers summing to 1
Posted:
Nov 17, 2012 11:52 AM


El viernes, 16 de noviembre de 2012 00:39:24 UTC4:30, CharlieBoo 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

