Andrzej Kukla @Mathinity_ Twitter profile

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Andrzej Kukla

@Mathinity_

4 months

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Andrzej Kukla

@Mathinity_

11 months

Even though I'm certain that this identity is true (3 proofs in this 🧵), I still can't believe it's true! [1/4]

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Andrzej Kukla

@Mathinity_

9 months

Here's another proof that the Fibonacci Sequence can appear literally in ANY problem. Enjoy the 🧵! [1/6]

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Andrzej Kukla

@Mathinity_

11 months

The number e, also called Napier's constant, is well known to be an irrational number. Despite the fame of this fact, not many people know how to prove it. It turns out to be surprisingly easy! You can find Joseph Fouriers proof below. [1/2]

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Andrzej Kukla

@Mathinity_

10 months

Here's a fun theorem I got to know lately. It's not only intriguing but also surprisingly easy to prove. Proof in the 🧵! [1/3]

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Andrzej Kukla

@Mathinity_

6 months

Here's a probability theory test I took yesterday. Can you do it in 30 minutes?

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Andrzej Kukla

@Mathinity_

8 months

In calculus we are often interested in checking whether a certain series converges or diverges. One of the most interesting ways is the Cauchy Condensation Test. Its name is quite literal. In this thread we prove it and then put it to use in a quite fun problem. Enjoy! [1/6]

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Andrzej Kukla

@Mathinity_

11 months

e is irrational, a proof [2/2]

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Andrzej Kukla

@Mathinity_

5 months

Happy New Year folks! 🥳 Share your own fun facts about the number 2024 😌

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Andrzej Kukla

@Mathinity_

10 months

The sum of first n odd numbers is equal to n². Classic. Here's a very necessary 🧵that consists of 7 different proofs of this fact. Let's get into it! [1/9]

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Andrzej Kukla

@Mathinity_

3 years

Imagine using FLT in your Algebra 1 class

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Andrzej Kukla

@Mathinity_

9 months

Sum of reciprocals of first prime numbers grows reaaally slowly (~ ln(ln(n) ), but nonetheless... it diverges! 🤯 Let's go over a beautiful and quite elementary proof, which is a small modification of the proof Ivan Niven. Enjoy this 🧵! [1/5]

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Andrzej Kukla

@Mathinity_

7 months

Legends say that the geometry fans are still waiting.

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Andrzej Kukla

@Mathinity_

11 months

3rd proof [4/4]

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Andrzej Kukla

@Mathinity_

11 months

Power of a point theorem is one of those results in geometry that immediately catch your eye with short and easy formulation and a close-to-magic result. Let us go over it's proof with Jakob Steiner, the man who introduced the concept of power of a point. [1/3]

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Andrzej Kukla

@Mathinity_

7 months

The sum of first n natural numbers is equal to n(n+1)/2. The sum of first n squares of natural numbers is equal n(n+1)(2n+1)/6. But did you know that the sum of m powers of first n natural numbers can be calculated using all the previous (m-1, m-2, ..., 0) sums? More info below.

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Andrzej Kukla

@Mathinity_

9 months

The Floor Function is one of the most basic functions one can think of, yet it's presence makes some problems very hard to tackle. In this 🧵 I want to go over a very simple trick that makes some problems more approachable. Let's get into it! [1/6]

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Andrzej Kukla

@Mathinity_

11 months

Did you know that the infamous L'Hôpital's rule has a discrete cousin, the Stolz–Cesàro theorem? Proof, remarks and an application can be found in the link below.

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Andrzej Kukla

@Mathinity_

6 months

Here are my solutions of that probability theory test I've shared recently. Enjoy the 🧵! [1/3]

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Andrzej Kukla

@Mathinity_

11 months

What is some particularly elegant proof you've read recently? For me it's the following proof of Hermite's identity. Delightful!

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Andrzej Kukla

@Mathinity_

9 months

Mathematical Induction is a mathematical tool that allows you to prove statements that involve natural numbers in a very slick way. And it is awful... ... kinda. [1/7]

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Andrzej Kukla

@Mathinity_

5 months

🎄❤️ Merry Christmas everyone! 🎄❤️ I'm waiting for all of your amazing solutions! 🙌

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Andrzej Kukla

@Mathinity_

11 months

1st proof [2/4]

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Andrzej Kukla

@Mathinity_

10 months

This peculiar series is very eye-catching and actually not that hard to evaluate! Give it a try! Also, it's the first Mathinity Sunday Challenge actually posted on Sunday in a long time!

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Andrzej Kukla

@Mathinity_

7 months

This math problem is so simple that it's "almost obvious". Yet it stumped me when I first tried to solve it and I believe that You might have some trouble with it too! Let a Triangle ∆ be inside a Rectangle ☐. Prove that Area(∆) ≤ ½Area(☐). Give it a try! 😉

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Andrzej Kukla

@Mathinity_

8 months

Classic calculus challenge with many different solutions. Be creative! 🤩

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Andrzej Kukla

@Mathinity_

9 months

This math problem is great for various reasons: - it is not that hard to solve, yet it is very satisfying and surprising, - it's a self made modification of a Putnam problem - and most importantly, it's a part of the IGMO2023 shortlist! Enjoy! 😊

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Andrzej Kukla

@Mathinity_

11 months

Clever test for whether a number is Fibonacci, stated and proved by Ira Gessel in 1972 (link to his proof below).

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Andrzej Kukla

@Mathinity_

7 months

This is a truly marvelous and devilish exercise. At first glance I thought it was a mistake that there are no assumptions about α and β. I was wrong. Good luck 😈

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Andrzej Kukla

@Mathinity_

8 months

In mathematics there are some problems for which it is hard to find a balance between simplicity and elegancy of the solution. In this thread I want to present a solution that could be compared to a sledgehammer cracking a nut. I hope you will find reading it enjoyable! [1/8]

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Andrzej Kukla

@Mathinity_

10 months

This Number Theory result is surprising on two levels - its statement is astonishing but, surprisingly, it's not that hard prove! Give it a go! Do you know the name of a theorem that generalizes it?

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Andrzej Kukla

@Mathinity_

5 months

Jay Cummings

@LongFormMath

5 months

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Andrzej Kukla

@Mathinity_

6 months

Mathematicians of the world, I need your help! What is the best Christmas present 🎄🎁 you can give to a mathematician?

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Andrzej Kukla

@Mathinity_

9 months

This problem is very special to me. Not only does it involve my favourite function, the floor function, but also it was created by me for the 3rd edition of IGMO, the math Olympiad I organize. Enjoy! 🌞

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Andrzej Kukla

@Mathinity_

8 months

Enjoy your 🥧!

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Andrzej Kukla

@Mathinity_

11 months

2nd proof [3/4]

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Andrzej Kukla

@Mathinity_

6 months

Calculating this sum is a classic problem I’ve personally encountered multiple times on various courses. I will show you 3 solutions which all have one thing in common - they all use The Angle Sum Identity. Enjoy the thread! 🧵 [1/]

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Andrzej Kukla

@Mathinity_

1 year

This is a 1.5 hours complex analysis exam I've taken today. Would you be able to solve all the problems in the intended time? 🤔

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Andrzej Kukla

@Mathinity_

10 months

Easy and satisfying identity that involves Fibonacci numbers! Warning, this 🧵 contains at least 3 different proofs of this fact! Also, if you know any different proof, please post it in a reply! Lets collect as many proofs in this 𝕏-tweet as possible! [1/5]

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Andrzej Kukla

@Mathinity_

6 months

Solution to the devilish 😈 exercise in mathematical logic. Btw should I keep posting handwritten stuff, or go back to LaTeX?

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Andrzej Kukla

@Mathinity_

6 months

Solution to the Riemann Hypothesis, duh.

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Andrzej Kukla

@Mathinity_

11 months

It is well known that the number of ways to tile 2 × n grid with 2 × 1 dominoes is equal to (n+1)'st Fibonacci number. What if we considered a 2 × n grid in a form of Möbius strip instead? Sarah-Marie Belcastro answers this question in her 2023 paper (link below).

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Andrzej Kukla

@Mathinity_

9 months

Easy, but quite enjoyable exercise. Give it a go and share your solution in replies!

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Andrzej Kukla

@Mathinity_

6 months

Combinatorics problem I absolutely adore. It ended up as the Q2 in the first round of Jagiellonian Mathematics Tournament, 7th edition! Enjoy!

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Andrzej Kukla

@Mathinity_

8 months

Here's how I turn ideas into solutions. Btw should I finally get a new whiteboard? Or maybe chalkboard would be better?

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Andrzej Kukla

@Mathinity_

5 months

Most obvious choice in calc 1

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Andrzej Kukla

@Mathinity_

8 months

Floor of x is defined as the largest integer smaller than or equal to x. Recently I’ve been thinking about a certain interpretation of the Floor function and it got me wondering... can we generalize it? Here's my journey towards the answer, in a form of a thread [1/8]

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Andrzej Kukla

@Mathinity_

11 months

Credit for the third proof goes to @ValenciaBats . Sorry about the late mention 🙏

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Andrzej Kukla

@Mathinity_

10 months

Entertaining series to evaluate! The solution itself is not particularly hard, but it requires a few non-trivial steps. All of them are contained in this 🧵 [1/5]

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Andrzej Kukla

@Mathinity_

11 months

I was given this exercise in my geometry class and I instantly fell in love with it. On one hand it's not at all trivial, but on the other hand there are so many ways to solve it!

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Andrzej Kukla

@Mathinity_

6 months

@Anthony_Bonato 3 proofs right here!

Andrzej Kukla

@Mathinity_

11 months

Even though I'm certain that this identity is true (3 proofs in this 🧵), I still can't believe it's true! [1/4]

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Andrzej Kukla

@Mathinity_

8 months

I'm really happy with how this problem turned out. Very satisfying!

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Andrzej Kukla

@Mathinity_

9 months

Even though I don't really like calculus, I find this problem truly gratifying. I hope you will enjoy this as much as I do 😌

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Andrzej Kukla

@Mathinity_

10 months

First part of the proof where we prove that the union of Spec(α) and Spec(β) contain all integers. Note that we define Spec(α) as {⌊α⌋, ⌊2α⌋, ⌊3α⌋, ...} [2/3]

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Andrzej Kukla

@Mathinity_

10 months

Second part of the proof where we prove that Spec(α) and Spec(β) are disjoint. [3/3]

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Andrzej Kukla

@Mathinity_

10 months

2nd proof, one of my favourites! [3/9]

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Andrzej Kukla

@Mathinity_

9 months

Final step aaaaaand... finish! 🤩 Thanks for reading, I hope you've enjoyed this thread! [5/5]

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Andrzej Kukla

@Mathinity_

6 months

@jeremiahjjohns It's a standard notation for random variables.

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Andrzej Kukla

@Mathinity_

2 years

I've completed my Bachelor's degree ❤️

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Andrzej Kukla

@Mathinity_

9 months

🚨 New paper alert 🚨 In this short thread we'll go through the results proved in "On Some Results on Practical Numbers" written by Sai Teja Somu, Ting Hon Stanford Li and myself, published yesterday in INTEGERS. [1/5]

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Andrzej Kukla

@Mathinity_

1 year

I really like this #analysis problem because it has a very surprising answer. Give it a go!

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Andrzej Kukla

@Mathinity_

1 year

#DiscreteMath question I fell in love with few days ago. Give it a go! #mathinity

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Andrzej Kukla

@Mathinity_

8 months

Here's a pleasing, highschool-level, geometry problem from the 2020 edition of IYMC, Pre-Final round. Enjoy!

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Andrzej Kukla

@Mathinity_

6 months

Solution to the latest, juicy 🧃 combinatorics problem. Split into two parts, but explained decently, at least I hope so. Enjoy! [1/2]

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Andrzej Kukla

@Mathinity_

10 months

There's nothing better than a simple and fun number theory exercise. Try it for yourself! By the way, Monday is the new Sunday everyone! 🙃

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Andrzej Kukla

@Mathinity_

5 months

Srinivasa Ramanujan was born on 22nd December in 1887. Happy 136th birthday math genius!🥳 In 2022 @TejaSai4 and myself have published a paper that continues the work of Ramanujan. In this paper we find and prove generalizations of Ramanujan Floor Function Identities. Link below

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Andrzej Kukla

@Mathinity_

1 year

One of my followers sent me this problem and it's pretty cool! Try it!

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Andrzej Kukla

@Mathinity_

9 months

First part, in which we sketch the whole proof [2/5]

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Andrzej Kukla

@Mathinity_

9 months

Modular arithmetic is the core of Number Theory and, even though it is used for many serious things, we are going to utilize it for a rather simple problem. Here's a 🧵 on how to prove that a perfect square cannot consist only of digits 0, 3, 5, 7 and 8. [1/6]

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Andrzej Kukla

@Mathinity_

10 months

1st, classic proof [2/9]

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Andrzej Kukla

@Mathinity_

11 months

@PhysGorg Actually Napier mentions 2.71... in his work almost a century before Euler was born. Euler indeed worked on it and gave it its name, "e".

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Andrzej Kukla

@Mathinity_

9 months

Proof of the second step. Almost there! [4/5]

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Andrzej Kukla

@Mathinity_

2 years

Yellow area = Blue area. Can you find x?

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Andrzej Kukla

@Mathinity_

9 months

Proof of the first step (+ the definition of square-free numbers) [3/5]

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Andrzej Kukla

@Mathinity_

9 months

Part 2 of the solution. If you've liked this 🧵, make sure to follow for more such threads in the future! [6/6]

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Andrzej Kukla

@Mathinity_

11 months

In the first place we are going to take care of the case when P is outside of the circle. [2/3]

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Andrzej Kukla

@Mathinity_

9 months

Floor function is my favourite function, mostly because of its complicated nature. Even though it is very simple, its presence in certain math problems makes them very hard to tackle. Thankfully, there is this almost trivial trick that makes some of them approachable. [3/6]

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Andrzej Kukla

@Mathinity_

7 months

@HassanMohyudDin That is not a valid proof.

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Andrzej Kukla

@Mathinity_

9 months

Now let’s go over a more insightful, algebraic proof. [4/8]

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Andrzej Kukla

@Mathinity_

11 months

Now we take care of the second case when P is inside the circle. [3/3]

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Andrzej Kukla

@Mathinity_

6 months

My favourite solution out of those 3! Do you agree? [3/3]

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Andrzej Kukla

@Mathinity_

10 months

3rd proof. Of course, induction had to be included [4/9]

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Andrzej Kukla

@Mathinity_

9 months

Part 1 of the solution. [5/6]

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Andrzej Kukla

@Mathinity_

10 months

6th proof, definitely my favourite one [7/9]

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Andrzej Kukla

@Mathinity_

9 months

Fibonacci sequence is one of the most famous sequences in the world. It is easy to understand and „mysterious” at the same time. Surprisingly it often appears in concepts and problems seemingly unrelated. In this thread we go through the following example of such problem. [2/6]

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Andrzej Kukla

@Mathinity_

9 months

The Floor Function or is one of the most basic functions one can think of. Essentially it is defined as the as the largest integer smaller than or equal to x. Analogously we define the fractional part function as x - floor(x). [2/6]

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Andrzej Kukla

@Mathinity_

7 months

Legends say that the measure theory fans are still falling. Yes, it's the same meme as before, but vol.2. Deal with it.

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Andrzej Kukla

@Mathinity_

8 months

This test is „condensing” the series we are checking the convergence of by only considering the 2^n indexes and compensating this reduction by the 2^n factor. Let’s prove this tests validity, shall we? [2/6]

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Andrzej Kukla

@Mathinity_

7 months

discrete math >>> math >>> calculus

Anthony Bonato

@Anthony_Bonato

7 months

Student: I prefer calculus over discrete mathematics. Me:

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Andrzej Kukla

@Mathinity_

1 year

I absolutely adore this problem. You gotta try this one out! #discretemath #math #numbertheory

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Andrzej Kukla

@Mathinity_

10 months

4th proof, second without words! [5/9]

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Andrzej Kukla

@Mathinity_

9 months

The main reason why I don’t quite like Mathematical Induction is that even though it can prove several statements, it tends to hide the insight about a problem that one might obtain by solving it in a different way. Let me demonstrate it with one of my favourite examples. [3/7]

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Andrzej Kukla

@Mathinity_

3 years

Really cool floor function problem! Try it out 👀

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Andrzej Kukla

@Mathinity_

10 months

5th proof, by means of discrete calculus! [6/9]

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Andrzej Kukla

@Mathinity_

5 months

Here goes... Nothing! The newest challenge is the outcome of my experiments with a very famous sequence of numbers that I happen to write my masters dissertation about. Can you guess what sequence it is? 🧐

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Andrzej Kukla

@Mathinity_

1 year

My prof (dr. prof UJ Krzysztof Ciesielski) shared this problem with me lately. It's amazing. Give it a go!

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Andrzej Kukla

@Mathinity_

9 months

Let's test this trick by solving the following problem. As always, try the problem before seeing my solution. Maybe you'll come up with a better solution than mine! [4/6]