What would you say are the prerequisites for this course? It looks interesting but I've never taken a Comp Arch course myself and worry I'll be too far out of depth.
Give it a try. The lectures I looked at don't look like they require much more than being able to read basic assembly language, and some of them probably don't even need that.
The GPU lecture is particularly good and doesn't seem to require much more than basic programming knowledge.
Maybe I'm being super pedantic and possibly confrontational (I apologise in advance) but it jumped out on me here. I think you've misunderstood the 'famous elementary proof' that there are infinite primes. You do not create a new prime as you have suggested (quoted below).
"Multiply them together and add one. No prime divides the new number (because "every" prime leaves a remainder 1), so you've just produced a new prime. This new prime is larger than the largest prime in your finite set because you multiplied that by the rest of them and added one to get it."
Instead you have created a number that may or may not be prime but definitely requires a new prime number (not in your set) to factorise it. Counter example: Take your set of prime numbers to be {2,3,5,7,11,13} then
(2.3.5.7.11.13)+1 = 30031
30031 factorises into 59.509 so you have found two prime numbers that are not in your original set.
EDIT: Responding to the edit above. The problem is that you claim that you make a new prime number by multiplying them all together and adding one. You didn't multiply all the numbers and added one to get the prime number, you multiplied all the numbers and added one to get a number (POSSIBLY NOT PRIME) whose FACTORS are prime numbers not in your original 'supposed' finite set. Your proof essentially lacks the step: IF new_number is prime: proof finished ELSE: factor new_number and show that at least one of the factors is not in your finite set.
EDIT 2: Counterexample number 2. Suppose your finite set of primes is {2,7}
(2.7)+1 = 15
So you have found 3 and 5 as primes that are not in your original set and are SMALLER than the largest prime in your original set. This is now a second mistake in your proof. Whether you are trolling or just too arrogant to see the mistake/error I do not know.
You're entirely correct. A very ironic mistake/misunderstanding for the parent post to make.
Edit: I've thought about this a bit more, and you can save the parent post by prepending a result like "Every number larger than one is either prime, or divisible by a prime smaller than itself". In this case you can then assert that your constructed number is prime. However, this result requires its own proof and was not mentioned by the parent post.
Technically, the parent post literally said "No prime divides the new number ... so you've just produced a new prime". It does not make the direct claim that the new prime is equal to the new number.
It is possible that the OP did not make the mistake that you are attributing to them, although best case is that the exposition was simply unclear.
Thanks, very charitable reading but doesn't work because next sentence I said that the new prime is larger than largest. As other poster points out if 2,7 are all your primes you produce 15, whose primes are not larger than 7 so the "new prime" wasn't referring to one of them.
But that's not how I argued. Instead I showed that 15 is the new prime because it's relatively prime to all the primes. (By the assumption we had started from all the primes.) And this is what falsifies the assumption.
I think my argument is fine and I don't need to address whether 15 is really a prime.
Your post says "No prime divides the new number, so you've just produced a new prime." What, exactly, do you mean by "produced"? You say produced, not "established the existence of", so it makes it sound like you are assuming a constructive argument.
Do you mean that the number you've produced is prime? Or do you mean that "the number you have computed is either prime or has a prime factors not in your list of primes, therefore, by computing the factorization of that number, you produce a new prime"?
The former statement is demonstrably false, the latter statement is true. Others have read your proof as stating the former, and I argue that the latter is a generous but not unreasonable interpretation of your phrasing.
Andrew, I perfectly see what you're saying, but I am telling you it is absolutely no problem that there's a false statement.
Remember that we are arguing from the false assumption that you start with the finite set of all primes, up to some largest prime, L. Call this false assumption A. It's fine to make false statements that follow from A, no problem at all. Indeed this is the point.
One definition of a prime is that there are no primes smaller than it in its prime factorization. We'll call this definition NSPIPF - no smaller prime in prime factorization. Is NSPIPF an OK test for primality? Sure.
So when you get to new_number, produce the prime factorization. Does it meet the definition of NSPIPF? Yes, because we made it relatively prime to every prime (under assumption A).
It passes primality test NSPIPF. Under A. And therefore is prime. Under A. (This is the part you and others object to, but it's absolutely flawless application of the NSPIPF primality test.) It doesn't matter that in some other way I could also get to a contradiction. For now this is what we do.
Under A, using NSPIPF primality test, we just proved new_number is prime.
Next we show that this new_number definitely wasn't in the set of all primes, since it's larger than L, having had L and a bunch of other positive integers as factors and then one added to that for good measure. It is at this point that we show explicitly that A cannot be true, because we used A to produce a prime that wasn't in the set of all primes.
It doesn't matter if that was a false statement!
I think all this is in my original comment and it is a flawless proof by contradiction. I asked you to "Suppose there are just finite primes, up to some largest" and you gave up when you started seeing false statements, rather than at the end of the paragraph where I presented the conclusion in black and white.
Nobody is contesting that the conclusion is true, just that the proof is unsound as written.
Here you've introduced a definition of prime that is different than the usual one, and is in fact self-recursive. NSPIPF is "no smaller prime in prime factorization". What is the first "prime" in this definition? Is it a number such that there is "no smaller prime in prime factorization"? What is the second "prime" in that definition? Is that the usual "prime factorization", or is it an NSPIPF factorization? In other words, how would you prove that 2 is prime given the NSPIPF definition?
It is more natural to talk about primality as a test that can be done independent of any assumptions about other primes, but rather as a matter of whether it can be expressed as the multiple of some number other than 1 and itself. That way we can make a precise statement about the product of the finite set of primes plus one without reference to the set of primes.
>It is more natural to talk about primality as a test that can be done independent of any assumptions about other primes, but rather as a matter of whether it can be expressed as the multiple of some number other than 1 and itself
This is clearly not what I was doing. I clearly referred to having no smaller prime factors. Anyway this aside is tiresome, it's like poking me for saying "every positive integer has a prime factorization" and then asking, okay, so what about 1 or something. I think my proof is fine and I'm not going to defend it anymore.
Suppose 2 and 7 are the only primes < 15. Then 15 is prime because its prime factorization couldn't contain any other number (2 * 7+1 makes it relatively prime to both 2 and 7 which under the supposition are the only primes < 15.) This is true if we suppose 2 and 7 are the only primes < 15, and that's just what the word "suppose" asks you to entertain.
This little mental exercise proves that 2 and 7 can't be the only primes in existence. It doesn't really matter what 15 really is or isn't. No need to add noise about it. We only care about the supposition, which you cut out of your quote.
Hope this explains why I didn't deal with the true status of new_number! It just doesn't matter.
However, I'm disappointed in you, in the other respondent, and in the moderation here, and I'm glad I made this alternate account.
Either disable JavaScript before visiting the Wayback Machine, or stop the loading of the page just after it has loaded the text but before it performs the redirection (a bit tricky, you have to stop it at the right time).
Given the link to the archive, it works without javascript for me. I disabled scripts from archive.org with umatrix and the archived page loaded just fine. The only difference from usual is the top bar that archive.org normally displays isn't present.
However you do seem to need javascript enabled to query the wayback machine from web.archive.org: "The Wayback Machine requires your browser to support JavaScript, please email info@archive.org if you have any questions about this. "
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