3. Initially, the numbers $1, 2, 3, \cdots, 2013$ are written on the blackboard. In each step, you can erase two numbers $a, b$ from the blackboard and write down $a+b$. What is the minimum number of steps required to ensure that the sum of any remaining numbers on the blackboard is not equal to 2014?
(Lin Chang provided the problem)
3. The minimum number of steps required is 503.

On one hand, in the 1006 pairs of positive integers
\[
(1,2013), \quad(2,2012), \quad \cdots, \quad(1006,1008)
\]
the sum of each pair is 2014, so at least one number in each pair must be operated on. Since each step connects two numbers with a “+” sign, by the principle of double counting, the number of operation steps is no less than \(\frac{1006}{2}=503\).

On the other hand, adding the two numbers in each of the 503 pairs of positive integers
\[
(1,1005),(2,1004), \cdots,(502,504),(503,1008)
\]
(totally 503 steps), the numbers on the blackboard become 503 instances of 1006, two instances of 1511, and one instance each of 1007, 1009, 1010, ..., 1510, 1512, 1513, ..., 2013. Each of these numbers is less than 2014, and the sum of any three of them is no less than \(3018 > 2014\), while the sum of any two of them can be \(2012, 2013, 2015, 2016, \cdots, 4025\). Therefore, such 503 steps of operation can meet the requirement.

In conclusion, the minimum number of operation steps required is 503.