cosecantphi [he/him, they/them]

nah, this is just the appetizer to a big bowl of pasta made out of antimatter.

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.

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.

I don’t think humans are capable of wiping out life on this planet for good. We’ll really mess shit up, there will be untold suffering amongst human and non-human sentient populations alike, but anthropogenic climate change is something that will take care of itself in short (on geological timescales) order once it causes our population to collapse. Life on Earth has survived much worse in the past.

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.

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.

This goes hard as fuck and I wish it were real.

I know this is going to make me sound like an insufferable asshole, but I’m kinda fucking sick of Uranus jokes, and would really like to change the planet’s name to be rid of them forever. I love astronomy and all that stuff, but you can’t talk about this planet without someone making the one joke about Uranus. The first few times we’re funny back when I was six years old and learning what a planet is. The next couple dozen times were pretty amusing as well, kids love butt jokes. But it’s time to stop. I’ve heard it all before, you’ve heard it all before, and no one has laughed at this joke in several decades at the least.

You know how you’ll click on a thread about an interesting topic that you’re looking forward to learn about, but no matter how far you scroll down all you can find is an endless horde of dipshit fucking redditors churning out the same assortment of the worst puns and references in the universe that they’ve been congratulating themselves about for the past twenty years? Uranus jokes are kind of like that, but worse.

Rename this planet George for all I care, I don’t give a shit. Just please end this nightmare.