The next number should be 13112221. Why?
Because each of these numbers describes the previous number. Starting with 1, the 2nd line is 11 (one one), then the 3rd line describes 11 as 21(two ones). Then the 4th line is 1211 which describes (one 2, one 1). And so forth...
Given two 2s, "plus" can be changed to "times" without changing the results: 2 + 2= 2 x 2.
The solution with three numbers is easy too: 1 + 2 + 3= 1 x 2 x 3.
Now find the answer for 4 numbers and for 5 numbers.
The word One has 3 letters, in the word three there are 5 letters, in the word five there are four letters and four has four letters in it as well. Whatever number you'll go back to four, and that's why it's the magic number.