Let r(a) = sqrt(a). Out of curiosity, I wrote a program to compute, for integer c >= 0 f(c) = (a, b) where integers a, b with 0 <= a <= b minimize r(a) + r(b) > r(c). For example, f(20) = (3, 8) because r(3) + r(8) is the smallest value of the form r(a) + r(b) > r(20). f is not quite well defined because f(24) = (1, 16) = (4, 9) so I proclaim that in ambiguous cases (this is the only one I encountered), smallest a wins, and f(24) = (1, 16). Anyway, I found something interesting. It turns out that f(10^2) = (13, 41) f(10^4) = (1337, 4024) f(10^6) = (236530, 263843) f(10^8) = (10806362, 45060299) There's nothing remarkable about these numbers until we look at their square roots: r(13) = 3.60555+ r(41) = 6.40312+ r(1337) = 36.56501059+ r(4024) = 63.43500610+ r(236530) = 486.34350000796+ r(263843) = 513.65650000754+ r(10806362) = 3287.30315000001178+ r(45060299) = 6712.69685000000577+ Whereas we might reasonably expect the digits to be random, we see curious runs of zeroes. This seems to be related to c being an even power of 10, but I have no explanation. I haven't tried other bases.