Are there two consecutive even numbers whose product is 224?

I want to find if there are two consecutive even integers whose product is 224, and if this can be generalized.

I saw this night before last and almost answered it, but got distracted and lost the question; that'll teach me to bookmark them!

Went out to resolved answers looking for it and found a great number of questions like it or, at least, containing 224; I have to assume that it is an unresolved question floating out there somewhere.

OK, let's set it up. M(M+2) = 224 , or

M^2 + 2M = 224.

Now bear with me; I'm going to digress, but you'll see where I'm going in a minute.

I noticed that 224 is one less than 15^2. This kind of thing stuck in my head when I was doing a lot of sum-and-difference of squares problems.

So if m = 14 (that's one less than 15) then 14^2 + 2x14 = 224....196 + 28 = 224.

So m = 14, and m+2 = 16; 14 x 16 = 224.

Although it's unrelated to the given problem, it seems obvious that there must be just such paired numbers around all the perfect squares.

For example m^2 + 2m = 624 (25^2-1),

which means that m = 24 and m+2 = 26;

24^2 +2x24 = 624

But just to be on the safe side, let's try it with the square of an even number, i.e. 900.

m^2 + 2m = (900-1) ; 29^2 + 2x29 =

841 + 58 = 899, so it would appear to work regardless.

Unfortunately, I don't quite know enough math to prove whether there are are numbers which do not fit this pattern; there may be.