Solution:

Let two consecutive odd numbers be 2k- 1 and 2k+ 1(k is a positive integer).

Then (2k+ 1)? -(2k- 1)？ =(4k？ +4k+ 1)-(4k？ -4k+ 1)=8k

That is, the so-called mysterious number must be a multiple of 8.

(1) 28 is not a multiple of 8, so it is not a mysterious number. 20 16=8x252 is a multiple of 8, so it is a mysterious number.

(2) It has been proved before, see the previous process.

(3) Let two consecutive even numbers be 2k, 2(k+ 1), then [2(k+ 1)]? -(2k)？ =8k+ 1, which is not a multiple of 8, so it is not a mysterious number.

[The word "even" in the title should be "odd"]