this is looking like some Escape from Dannemora shit lmao

hope these guys who escaped are safe out there, how likely is it they never get caught?

You're very welcome! I'm slowly teaching myself what is essentially a mathematics undergrad degree, and I'm familiar with the book you're using, so if you ever have any other questions feel free to ask!

Word of advice: In most math books, especially at lower levels, the majority of the exercises will nearly always be some sort of direct application of the theorems and proofs in the preceding chapter of instruction.

So when you feel stuck, you should go back to those theorems and try to make sure you really understand what the proof is saying. Like try to be skeptical about the statements and examine them until you are fully convinced that the proof is ironclad. Then it'll be much easier to spot which theorems each exercise is meant to provide elaboration/nuance on, especially the earlier exercises in a chapter.

The later exercises in a chapter tend to be much more difficult, but you'll nearly always be able to prove them with the theorems you've already learned, it's just that the harder ones will be essentially foreshadowing theorems in future chapters. So the longer you stick with this, deeply examining every theorem and attempting every exercise, the easier it becomes as you begin to understand the pedagogical intent of the author.

The "go outside and touch grass with friends and family on new years instead of being on hexbear" struggle session we had a few years ago really sucked

Take a closer look at the proof and conditions of conjecture 2 in chapter 1. It states that if you have a positive integer n that isn't prime (i.e it is n=ab for positive integers a and b both less than n, then the integer given by (2^{n)-1 = (2}ab) - 1 is not prime either.

But the proof itself for that conjecture gives you a means of computing integer factors x and y of any such number where n is not prime. It uses the telescoping property of sums to prove in general under these conditions that:

(2^{n) - 1 = ( (2}b) -1 ) * y

That is, one of the two factors you are looking for takes the form x = (2^b) - 1.

So let's use this to solve part (a) as an example. (2^15) -1 = 32767 is not prime according to conjecture 2 because 15=3*5, a product of positive integers less than n=15 . Now plug it into the equation with a = 3 and b = 5:

( ( 2^{3*5 ) - 1 ) = ( (2}5) - 1 ) * y

Now you just solve for y:

y = ( ( 2^{3*5 ) - 1 ) / ( ( 2}5 )- 1 ) = 1057

And we already had x = (2^5) -1 = 31

We may now easily confirm the result by multiplying: 1057 * 31 = 32767.

To apply this to part b, all that remains is to repeat the process with a = 31 and b = 1057.

Ok, we started with not voting makes you as bad as the literal fascists.

Now we're at voting doesn't materially effect the outcome in the vast majority of cases, but you're still as bad as the fascists for not participating in the American civil religion by personally legitimizing our deeply undemocratic institutions with a vote that is essentially an endorsement of further neoliberal policy that would gladly see many of us dead or enslaved.

Is politics just like a sports thing to you or something? Where you show up to support your team every couple years? Why should I care if the Democrats or Republicans are in office if they are both committing genocide, if they both support policy that is a direct threat to the lives and livelihoods of myself and people I care about? I don't want to be complicit in that, but apparently I'm as bad as the fascists for not voting for actual fascists in sham elections?

Voter opinion doesn't correlate to policy in literally any study I've ever seen of the electorate in this genocidal farce of a settler-slaver colony. The problem is very clearly not a lack of voter participation, but a lack of actual, functioning democratic institutions. You're never gonna vote your way out of this situation because the capitalists built the entire system to prevent you from doing exactly that.

When you make a big show of voting blue no matter what, you tell the Democrats they don't need to worry about your interests because they have your vote regardless. Why the fuck would you do that?

I had that happen to me a couple times on freebase DXM but never on HBr. Was it a large dose, high plateau trip?

Tripping on DXM. This drug is the OTC poison of choice for desperate teenagers, and that unfortunately gives it a very bad reputation.

But DXM is actually one of the safer hallucinogens out there so long as you get a DXM only product and don't mix it with certain other drugs. It's definitely the closest thing to Ketamine you can buy at walmart or amazon, that's for sure. Highly recommend that one.

edit: read the title backwards on this one, this gets more hate than it deserves haha

It's true! I did this one a few years ago after hearing about it on Hexbear

https://hexbear.net/post/106725?scrollToComments=false

No, I read what you wrote. I am upset about exactly what you wrote. I want you to listen to trans people and their experience. I want you to realize your experience as a cis person makes your opinions irrelevant on the matter and if you want to be a good ally you need to purge transphobia from the spaces you moderate.

Why should we engage in good faith on your "ideas" of wanting to fucking murder us with your bigotry? Unironically, the world would be better off without you.

It honestly sounds a lot like Hexbear. Made a rednote account today, but I've never used tiktok or instagram before, so the entire thing was kinda inscrutable for me lol, but I'm so glad that this is where the tiktok exodus is heading.

Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences.

Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process.

That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis.

So we have:

0.101001000100001000001 . . . = 10^{-1 + 10}-2 + 10^{-3 + 10}-4 +10^-5+ . . .

Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle.

Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it.

It's implicitly defined here by its decimal form:

0.101001000100001000001 . . .

The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating.

One way it could make sense is if the Adjuster initially intended to lay low and move on, but once he saw how positive the social media reception was he decided he'd take credit for it and try his luck convincing one of twelve jurors to nullify.

It sucks to realize in hindsight, but if Bernie had won the 2016 primary he would have been thoroughly ratfucked as a matter of bipartisan consensus. The Democrats would have sabotaged his national campaign at every step, probably even going as far as to have a few high profile Democrats endorse a third party candidate to make sure Bernie loses.

The reason you need to slow down is because you're starting on Earth, which means you're moving fast enough parallel to the sun's surface that for every foot you fall downwards toward the sun, the sun's surface curves away by 1 foot. This results in the nearly circular orbit around the sun we exist in.

If you start speeding up, the orbit becomes more elliptical, except your aphelion starts raising away from the sun because now you're moving fast enough that you've moved more than 1 foot sideways in the time you've fallen 1 foot downwards.

Slowing down has the opposite effect. If you get your speed down to 0, you'll fall straight down toward the sun as normal with gravity. But you don't need to go all the way down to 0 velocity to enter the sun, you just need to slow down until your elliptical orbit brushes up against the sun's surface. If you then want to speed back up to avoid falling into the sun, you need to do it parallel to the sun's surface. At this point, speeding up toward the sun will actually make you fall into the sun faster.

So basically the problem isn't that you're moving too fast to fall into the sun. By virtue of Earth's orbit, you're moving too fast in a direction away from hitting the sun's surface.

Actually childish understanding of geopolitics and history