Sum of Square Roots Problem Consider these two lists A = {1,25,31,84,87,134,158,182,198} B = {2,18,42,66,113,116,169,175,199} Now for both lists, add 1,000,000 to each of the numbers then take the sum of their square roots. For example, list A gives us $\sqrt{1000001}+\sqrt{1000025}+...\sqrt{1000198} = 9000.44998...$ Now what is interesting to note here is that the number obtained from list A and the number obtained from list B differ first in the 37th decimal place. A gives us the digit 2 and B gives us 5. Any explanation as to how/why does this happen? Or rather how were these numbers chosen? Is there some theory behind this, an algorithm perhaps to compile such lists or is it just some random number theory abomination?

A quick check shows that $$\sum_{a \in A} a^k = \sum_{b \in B} b^k$$ for $k \in \\{1,2,\ldots,8\\}$. So you can approximate it using binomial theorem $$\sqrt{a+x} = \sqrt{a} \sqrt{1+\dfrac{x}a} = \sqrt{a} \times \left(1 + \dfrac12 \dfrac{x}a - \dfrac12 \times \dfrac12 \left(\dfrac{x}{a} \right)^2 + \dfrac12 \times \dfrac12 \times \dfrac32 \left(\dfrac{x}{a} \right)^3 \mp \cdots \right)$$ to see that the difference between the two is of the order of $$1000 \times \dfrac12 \times \dfrac12 \times \dfrac32 \times \dfrac52 \times \dfrac72 \times \dfrac92 \times \dfrac{11}2 \times \dfrac{13}2 \times \dfrac{15}2 \times \left(\dfrac{\sum_{a \in A} a^9 - \sum_{b \in B} b^9}{1000000^9} \right) \approx 10^{-37}$$ (since the first nine terms in both the expansions are the same).