this post was submitted on 25 Sep 2026
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I'm not confused. I've been teaching this subject for over a decade.
I'm not certain are you arguing with me?
I was largely agreeing with you.
Four decades ago, the professor who taught me P vs NP absolutely stuffed his explanation of the polynomial refactoring example, drawing sum of products on the board as his example - bugged the crap out of me because I could visualize code for an algorithm to refactor sum of products relatively easily - certainly in polynomial time. I brought that to him after the next lecture and he clarified: product of sums is the hard problem, well, yeah, obviously when you look at that nightmare. I assume he had been teaching it for about a decade as well, he certainly had been in the department that long and longer.
You wrote "we know for certain do that it [factoring] is not in NP hard". That sentence is 1) borderline ungrammatical (we usually use NP-hard as an adjective, though it also denotes a set); and 2) in error. We suspect factoring is not NP-hard but we don't know for certain (consider what happens if P=NP). The error suggests confusion about what these terms mean. If you're really teaching this subject, can I ask what textbook you are using?
No, we don't use it as an adjective. It is a set. That's literally all it is.
And predicating anything on the assumption that P equals NP is rather absurd. Of course, we can't rule it out as a possibility. But virtually nobody in the discipline believes that to be the case. And honestly, why would it be? If there were polynomial time solutions to that many problems of that degree of importance, surely we would have discovered something by now.
While I agree that NP is likely not within P, that "No True Scotsman" argument is exceptionally weak for such a well disciplined field.
From the first sentence of https://en.wikipedia.org/wiki/NP-hardness : "In computational complexity theory, a computational problem H is called NP-hard if...".
Exactly, we don't know for certain. Of course it's very unlikely, but in math when we say we know something for certain, it means there's a theorem to that effect. There reason to think that factoring is not NP-hard but we don't know for certain. If you still claim otherwise, can you cite a theorem?
Anyway, yes, you're confused, and at this point you're spouting misinformation. You might consider reading a book or taking a class.
It's still an open problem, there's a $1 million Clay prize waiting for you to claim it if you have a proof either way.