Problem A Is Harder Than Problem B
There are times when we don't really know how hard a problem is (whatever that might mean) but we do have a sense that one problem is harder than another. We might not know the absolute difficulty, but we might have a concept of the relative difficulty.
To help gain some intuition, here is a specific example ...
So 1111111 is not prime. Our ability to solve instances of problem F gives us the ability to solve instances of problem P. It's simple to see that this always works, and thus we have shown that factorising is at least as hard as determining whether a number is prime.
Well, firstly, even though it's something of a mouthful, it's easiest to deal with the relationship "At least as hard as". This then allows that two problems are considered to be of equal difficulty. Even so, what does it mean to say that Problem A "is at least as hard as" Problem B?
We proceed on the idea that if problem A is harder (or at least as hard), then once we've solved problem A, solving problem B should be easier (or no harder).
So let's suppose we can solve problem A.
If we can now show that with a small amount of work we can convert an instance of problem B into an instance of problem A, and if we can also show that with another small amount of work that solution to that instance can then be converted back to a solution of the original instance, then that means that with a small amount of work we can solve B. We convert an instance of B to an instance of A, solve it, and then convert the solution back.
If we can do that, then assuming that we can solve problem A automatically implies that (with a "trivial" amount of work) we can solve B as well.
In that case we would have to consider problem B to be no harder than problem A.
In the example we've been assuming that factoring is strictly harder than primality testing (unless P=NP). We have efficient ways to test if a number is prime, but we don't have efficient ways to factor. As a specific example of that, we know for sure that this number:
7403756347956171282804679609742957314259 3188889231289084936232638972765034028266 2768919964196251178439958943305021275853 7011896809828673317327310893090055250511 6877063299072396380786710086096962537934 650563796359is not prime, but no one (at the time of writing) knows its factors. You can find it and others here: