Question

My prof gave an alternative explanation for checking till sqrt(n) by replying this:

“We need to check for divisors only till max_{2 ≤ i ≤ n}(min(i, n/i)).
The maximum is attained when i = n/i, ie. i = sqrt n.”

What does he even mean? Can someone put it in English?

Answer 1

Formatting the TeX for you, that’s “max_{2 <= i <= n}(min(i, n/i))”. In English, the largest value of the smaller of i and n/i over all values of i from 2 up to n.

For example, if n is 12:

i    n/i    min(i,n/i)
2    6      2
3    4      3    <--- Largest value is 3: sqrt(12) rounded down
4    3      3
5    2      2
6    2      2
7    1      1
8    1      1
9    1      1
10   1      1
11   1      1 
12   1      1

It’s quite easy to see that i < n/i if and only if i < sqrt(n), and from that we can see that the largest value of that expression will be sqrt(n).

Presumably, this is for finding the factors of n. If i is a factor, then n/i also is, since n/i * i = n, so there’s no need to test both i and n/i. Therefore, we can choose only to check the smaller of the two, min(i, n/i), and only need consider values of i up to the largest such value – which is the value of the expression your teacher gave you.