xyymath

[xyymath]

Graduate school is the art of turning ‘I think’ into ‘As supported by the evidence, it is reasonable to propose.’

[xyymath]

is 0 an even number?

Oh yes. 0 is even. Definitely even. By definiton.

Even numbers are multiples of 2, yes?

which means it can be expressed in the form of 2k where k is an integer. 0 = 2 x 0, where k = 0 , an integer

why isn't it odd?

because an odd number is of the form 2k + 1, where again k is an integer

so let's assume 0 is an odd number, then

0 = 2k + 1

which gives : k = -1/2

but -1/2 is not an integer.

therefore 0 is not an odd number (proof by contradiction)

So there you go. 0 is an even number. Needless to say, there are various other ways to go around proving that 0 is an even number. But this ig, is the easiest to go by.

@1sundriedtomato1

[xyymath]

You know you're in academia when you’ve spent more time figuring out your citation style than actually researching the topic.

[xyymath]

‘The data speaks for itself’ is the biggest lie in academia. The data mumbles, and you must be its translator.

[xyymath]

College: Where you learn that ‘free time’ is just a myth perpetuated by people who aren’t enrolled in your classes.

[xyymath]

When you spend more time formatting your paper than actually writing it, you know you’re in academia.

[xyymath]

Why Are Twin Primes So Rare? (And What Makes Them So Special)

When it comes to prime numbers, one of the most fascinating—and still unsolved—problems is the Twin Prime Conjecture. This conjecture centers on prime pairs that are only two units apart, like (3, 5), (11, 13), and (17, 19). These pairs are known as twin primes, and while they seem to pop up pretty regularly at smaller numbers, the farther you go into the number line, the less frequent they become. The conjecture suggests there are infinitely many twin primes, but... proving it? That’s a whole other beast.

To understand why these prime pairs are so rare, let’s dive into some prime distribution basics.

The Prime Number Theorem (PNT): Where Do Primes Hide?

At the core of understanding why twin primes are so elusive is the Prime Number Theorem (PNT). PNT gives us a way to estimate how many primes exist below a certain number x. In simple terms, it tells us that the number of primes up to x is roughly x/lnx​. In other words, primes get scarcer as we get higher into the number line.

But here’s the kicker: The gaps between primes aren’t exactly consistent. They fluctuate. Some stretches have big gaps, while others have small ones. That’s where twin primes come in—they’re a prime pair that’s so close together, just two units apart. The problem, though, is that while primes are becoming more spaced out as numbers increase, these close-knit pairs don’t show up as often.

So, What’s the Conjecture All About?

The Twin Prime Conjecture basically says that there are infinitely many primes p such that both p and p+2 are prime. But here's the thing: The further we go, the rarer these twin primes seem to get. You’ll still spot a few here and there, but can we really expect an endless stream of them? That’s the heart of the conjecture—and why it’s so tricky to prove.

Here’s why: The prime gap (the difference between two consecutive primes) gets bigger over time, but it doesn’t follow a clean, predictable pattern. Sometimes primes are close together, and sometimes they’re far apart. Thanks to sieve theory and some deep analytic number theory, we know that prime gaps generally grow as ∼(ln(x))^2, but they fluctuate. So, even though primes are becoming sparser, we can still find the occasional cluster, like twin primes.

The Big Breakthrough: Yitang Zhang’s Game-Changer

Now, for a little history lesson: In 2013, mathematician Yitang Zhang shook up the prime world with a major result. He proved that there are infinitely many pairs of primes that differ by no more than 70 million. This wasn’t quite the Twin Prime Conjecture, but it was a huge step forward. It essentially showed that the gaps between primes aren’t as large as we once thought. While the gap of 70 million is far from 2, it opened up new avenues for proving that there are infinitely many twin primes.

Zhang’s approach used some advanced techniques, like selberg sieve methods and modular forms, to bound the gaps between primes. This work significantly reduced the upper bound for prime gaps, giving us hope that someday we might tighten the gap even more and, who knows, prove the Twin Prime Conjecture itself.

What’s Next?

The Hardy-Littlewood conjecture, which is related to twin primes, predicts that the number of twin primes up to a given number x behaves like C⋅x/(ln⁡(x))^2 ​, where C is a constant. This suggests that twin primes aren’t as rare as once thought—though figuring out exactly how often they appear is still a bit of a mystery.

[xyymath]

How to say...

(Academic version)

1. “I have no idea.”

"Further investigation is required to draw a definitive conclusion."

2. “This is obvious.”

"It is evident from the data that..."

3. “This doesn’t work.”

"The proposed methodology yielded results inconsistent with the hypothesis."

4. “We need more money for this.”

"Further progress in this field is contingent upon securing additional funding."

5. “I’m guessing.”

"Based on preliminary observations, one might reasonably infer..."

6. “Someone else figured this out already.”

"This builds upon prior research conducted by [citation]."

7. “Nobody agrees on this.”

"This remains a contested topic within the field."

8. “I didn’t read the whole thing.”

"This paper was selectively reviewed for its relevance to the current study."

9. “This is wrong.”

"The results diverge significantly from established models, indicating potential inaccuracies in the assumptions."

10. “It’s complicated.”

"This phenomenon is influenced by a confluence of multifaceted variables."

11. “I got lucky.”

"The results were obtained serendipitously during the course of the experiment."

12. “This paper is awful.”

"The conclusions presented are ambitious and warrant cautious interpretation."

13. “This isn’t my problem.”

"Addressing this issue lies beyond the scope of the current study."

14. “I have no clue what they’re talking about.”

"The authors present an innovative but highly complex framework that necessitates further clarification."

15. “I don’t want to deal with this.”

"This issue remains an open question for future research."

16. “This was obvious before we even started.”

"The findings align closely with initial expectations, reinforcing established theories."

17. “This has nothing to do with my work.”

"While tangential to the primary focus, this topic offers intriguing avenues for supplementary analysis."

18. “This is boring.”

"The findings are methodologically sound but lack immediate groundbreaking implications."

19. “This idea is stolen.”

"This closely parallels earlier work by [citation], raising questions of originality."

20. “We need to fix this mess.”

"A revised methodology may be required to address the inconsistencies observed."

21. “This was a last-minute addition.”

"In response to recent developments, this aspect was incorporated to enhance the study’s comprehensiveness."

22. “This data is useless.”

"The dataset yielded limited actionable insights."

23. “I don’t want to write this section.”

"Details on this aspect are beyond the scope of this paper but can be explored in supplementary materials."

24. “I’m done with this.”

"In conclusion, this study represents a comprehensive effort to address the research questions, though further inquiries remain."

[xyymath]

Grad school: Where you learn the art of pretending you understand what your advisor is saying.

[xyymath]

‘This paper addresses a gap in the literature’ = ‘No one else thought this was worth investigating.’

[xyymath]

The academic pyramid: Publish or perish, cite or vanish, critique or be critiqued.

[xyymath]

Academia: where a single footnote can take three weeks to finalize.

[xyymath]

Submitting your paper: the intellectual equivalent of jumping out of a plane and hoping you packed a parachute.

[xyymath]

Academia is the only place where the phrase ‘further research is needed’ translates to ‘we’re just as confused as you.’

[xyymath]

One page into writing my paper: ‘I am a genius.’ Six pages in: ‘I am an imposter.’

[xyymath]

in case anyone had questions this was the movie this I was referring to:

Reviewing the Monty Hall Problem. Again. (Because my friends showed me that scene from 21. You know the one.)

If you’re unfamiliar, the Monty Hall problem is a probability puzzle based on an old game show. Here's how it works:

You’re presented with three doors. Behind one is a shiny car (yay!), and behind the other two are goats (meh).

You pick a door.

The host, who knows what’s behind each door, opens one of the other two doors, revealing a goat.

Now, you’re given a choice: stay with your original door or switch to the other unopened door.

[xyymath]

Proof 2 : using Conditional Probability (Bayes' Theorem)

Bayes' theorem states:

P(A|B) = {P(B|A) x P(A)}/P(B)

Where:

P(A): The prior probability of event A (e.g., picking the car).

P(B∣A): The likelihood of observing B given A

P(B): The total probability of observing B.

so in order to use this proof, we'll need to define certain events. I'm taking them as : (it's A subscript 1,2,3 actually)

A1​: You initially pick the car (Door 1).

A2: You initially pick a goat behind Door 2.

A3​: You initially pick a goat behind Door 3.

B: Monty opens a door with a goat.

We want to calculate P(A1∣B)(probability you win by sticking) and P(A2∣B), P(A3∣B) (probabilities you win by switching).

before plugging in the values in Bayes' formula, let's first calculate them, individually :

P(A1)=P(A2)=P(A3)=1/3 (equal probability of initially choosing any door).

P(B∣A1​)=1 (If you pick the car, Monty always opens a goat door)

P(B∣A2​)=1 (If you pick Goat 2, Monty opens Goat 3’s door.)

P(B∣A3​)=1 (If you pick Goat 3, Monty opens Goat 2’s door.)

So the total probability P(B) is:

P(B) = P(B∣A1​)⋅P(A1​) + P(B∣A2​)⋅P(A2​) + P(B∣A3​)⋅P(A3​) = 1

Now when we calculate we get these values:

P(A1|B) = P(A2|B) = P(A3|B) = 1/3

If you stay with your initial choice, you win only if A1​ is true (you initially picked the car). Probability of winning = P(A1∣B) = 1/3

If you switch, you win if A2​ or A3​ is true (you initially picked a goat). Probability of winning = P(A2∣B)+P(A3∣B) = 1/3 + 1/3 = 2/3

So in conclusion : switching gives you a 2/3 chance of winning, while staying only gives you a 1/3 chance. It’s a mathematically sound reason to always switch!

Reviewing the Monty Hall Problem. Again. (Because my friends showed me that scene from 21. You know the one.)

If you’re unfamiliar, the Monty Hall problem is a probability puzzle based on an old game show. Here's how it works:

You’re presented with three doors. Behind one is a shiny car (yay!), and behind the other two are goats (meh).

You pick a door.

The host, who knows what’s behind each door, opens one of the other two doors, revealing a goat.

Now, you’re given a choice: stay with your original door or switch to the other unopened door.