On partitions into squares of distinct integers whose reciprocals sum to 1

In 1963, Graham [1] proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured [2, Section D11] that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of such representations. Apositive integerm is called representable if there exists a set of positive integersX = {x1, x2, . . . , xk} such that 1 = 1 x1 + · · · + 1 xn and m = x21 + · · · + x 2 n.