Christian

If you want to show there are infinitely many primes, one way is to first note that every integer greater than 1 has a prime factor. This is because if an integer n is prime, n is a prime factor of itself, and if n is not prime then it must have a smaller factor m other than 1, 1< m < n. If m is also not prime, it too must have a smaller factor other than 1, and you can keep playing this game but there are only so many integers between 1 and n so eventually you’ll get to a factor of n that has no smaller factors of its own other than 1, which means it is prime.

Let’s now suppose there is only a finite number of primes, we’ll try to show that this assumption leads to nonsense so can’t be possible.

We can multiply any finite number of integers together to get a new integer. Let’s multiply all of the primes together to get a new number M. Then M + 1 gives a remainder of 1 when you divide by any prime number. Since dividing by a factor will always give a remainder of 0, none of the prime numbers can be a factor of M + 1. So M + 1 is an imteger bigger than 1 with no prime factors. This is impossible, so there must be a mistake somewhere in this argument.

The only thing we said that we’re not 100% sure is true was that there are a finite number of primes, so that has to be our mistake. So there must be infinitely many prime numbers.

I misread that as Radeon 9700 for a second and thought I had jumped back in time twenty years.

There are only finitely many prime numbers and I will not hear otherwise.

I don’t remember where I read this quote originally and I can only paraphrase it, but observing people living in a capitalist society and concluding that human nature is self-centeredness and greed is equivalent to observing workers in a factory that is poisoning their lungs and concluding that human nature is to cough.

I barely remember this anymore but the downgrade had certain things deactivated. Something like my card had four “pipelines” and the high-end one had eight, so a minor hardware modification could reactivate them. It was risky though, because often imperfections came out of the manufacturing process, and then they would just deactivate the problem areas and turn it into a lower-end version.

After a little while, someone put out drivers that could simulate the modification without physically touching the card. You’d read about softmod and hardmod for the lower-end radeon cards.

I used the softmod and 90% of the time it worked perfectly, but there was definitely an issue where some textures in certain games would have weird artifacting in a checkerboard pattern. If I disabled the softmod the artifacting wouldn’t happen.