I'm not a mathematician. I was recently watching a YT math question on what is 10! ÷ 6! = x!. It got me thinking, are there infinite triplets (a, b, c) that satisfy a!= b!⋅c!?
I wrote a short python program and found that there are many sets with (a+1, a, b) and you can construct them from (6, 5, 3) set onwards. Next set is (24 (6⋅4), 23, 4), the next set is (120 (24⋅5), 119, 6) and next is (720 (6⋅120), 719, 7), and so on. I think it is explained in this SE post. This is I think what mathematician would call trivial solutions.
In the mix I also found one interesting set (10,7,6). While is kind of similar with b = c + 1, it is still different from rest. And in the small program I wrote and with the resources I had, I could only find 1 such set below 1000. My question is, are there other such sets and are they infinitely many?
I tried google search but I think I'm not wording it properly that it brings. Would appreciate your inputs in pointing me in the right direction.