Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p2 must also divide n.
(8,9) is a couple of two consecutive numbers,both of them being powerful.
Find another pair(s) like that.
The more the merrier!!
(In reply to re: An interesting remark ***** Spoiler
1.Of course, it is obvious. That's why the sentence begins:...it is easy to see...etc
2.It is not "my discovery" - I've quoted the source.
3. There is a small typo in the 4th row of your post, which should read: . " .now it remains to show that 4n(n+1)+1
is also a p.n. "
You may still edit it.