Factoring is at most NP-complete because we have a polynomial time verification for it.
That means it’s in NP. These terms mean very precise things and it’s easy to get confused, but at the end of the day the new paper didn’t find a faster way to factor.
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.
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.
If there were polynomial time solutions to that many problems of that degree of importance, surely we would have discovered something by now.
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.
It’s likely to be NP-intermediate (outside of P, but not NP-hard), but it is not known.
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That means it’s in NP. These terms mean very precise things and it’s easy to get confused, but at the end of the day the new paper didn’t find a faster way to factor.
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.
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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.
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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.
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