Joseph Liouville and his Numbers (especially his constant)

Number theory. Whoo!

In number theory, let there be a number x where, for any positive integer n, there are infinitely many number pairs (p, q) where q > 1 that:

0 < |x – p/q| < 1/q^n

x is the Liouville number.

Liouville numbers can be described as “almost rational”, and can be approximated by sequences of rational numbers. In other words, Liouville numbers are transcendentals that can be more closely approximated than any other algebraic irrational.

In 1844, Joseph Liouville proved that all Liouville numbers were transcendental. Maybe, if I can find the proof, I might post why, but not now.

One Liouville number is the Liouville Constant, which takes the form of:

 L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001...
0.1100010000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001… I’m not sure if that’s 120 decimal digits.

One thing: Liouville’s Constant (call it L; TapL hahaha) almost satisfies 10x^6 -75x^3 – 190x + 21 = 0, if the solution was rounded, then maybe.

If my mind comes to it, I might go into more detail

RSA Factoring Challenge

The RSA Factoring Challenge is a set of factorisation challenges that involve numbers with hundreds of digits. The highest one cracked was RSA-250, which was 250 digits long, 829 bits, and was factored in February 2020 by Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé, and Paul Zimmermann (what’s with the cool names?). The number was:

2140324650240744961264423072839333563008614715144755017797754920881418023447140136643345519095804679610992851872470914587687396261921557363047454770520805119056493106687691590019759405693457452230589325976697471681738069364894699871578494975937497937

The factorisation was:

6413528947707158027879019017057738908482501474294344720811685963202453234463 0238623598752668347708737661925585694639798853367 × 3337202759497815655622601060535511422794076034476755466678452098702384172921 0037080257448673296881877565718986258036932062711

According to Wikipedia, the factorisation process utilised approximately 2700 CPU core-years, using a 2.1Ghz Intel Xeon Gold 6130 CPU as a reference. The program used to factorise RSA-250 was the Number Field Sieve algorithm, using the open source CADO-NFS software.

The dedication of the factorisation was towards Peter Montgomery, an American mathematician known for his contributions toward computational number theory and cryptography. He passed away on February 18, 2020.

The highest unsolved RSA Factoring Challenge is RSA-2048, with a whopping 617-digit number (2048 bits), with the number being:

25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357

And if the factorisation was found, the prize would have been US$200,000 but the challenge ended before anyone was awarded the prize. RSA-2048 may not be factorable, unless in the future, considerable advances in integer factorisation or computational power are made.

There are many other challenges, like RSA-617 with 617 digits, which still has not been factored, and the smallest RSA challenge number is RSA-260, which is:

22112825529529666435281085255026230927612089502470015394413748319128822941402001986512729726569746599085900330031400051170742204560859276357953757185954298838958709229238491006703034124620545784566413664540684214361293017694020846391065875914794251435144458199

So, if RSA-260 is cracked, I will be posting about it.

I think tomorrow, I’ll do the Riemann Hypothesis if I have enough time.

Which is closer to 16: 32 or 17?

In a +1 way, it’s 17.

But in a PRIME FACTOR way, it’s 32.

Why? And which is it?

  • Kindergarten kids: 17
  • Primary: 17
  • Middle School: 17
  • High school: 17
  • College: probably both, but more likely 17
  • University: 15? 17? 32? WHAT?!?!?!?!
  • Real-life mathematicians who only care about supernaturals: 32

(screech sound)

What on Earth is a SuPeRnAtUrAl?!?!?!?!?!?!?!?!

If you watch Vi Hart’s video “How many types of infinity are there?” at 9:04, it says “supernatural” numbers. You know indices, x to the power of y? if y is infinity, for any real x bigger than 1 will be infinite. The +1 definition sees 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*… the same as 7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*7*…, but in the supernaturals, the 7*7*7*7*7*7*7*7*7*7*… is bigger, because they are multiplied the same infinite amount, and every time, 2<7, 4<49, etc.

So in the supernatural way, the number 16 is closer to 32 than 17 because 16 and 32 are just times 2 away, while 17 is… well, does not have a common factor at all.

That’s good!

Q.E.D.

Happy 3.141592653589793238462643385279… day!

It’s πi day today!

(not to be confused with π times i, in which it would be, like, an imaginary day.)

Sorry I wrote this so late. I didn’t realize it was 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440… day, so forgive me, will you?

Will you forgive me for …

Remus Lupin, Harry Potter and the Prisoner of Azkaban

But this post is wrong (see Dear Archimedes,), and I will make a post on June 28 aka Tau Day!

Farewell, my friends (excluding the pi-fans)!

Gwang Ho Kim, the author (or Wanttolearncalculus)

Well, e.

Last post, I talked about e being used in other stuff. Well, the first example is this!!!

e^ia = cos a +isin a.

What?!?!?!?! That basically means that e to the power of sqrt(-1)*a (an angle expressed in radians/angles (but be sure to include the degree sign!)) equals the cosine of a plus sqrt(-1)*sin a. SEARCH IT UP, IF YOU DARE!!!

This is an important formula used in a lot of stuff. Even Euler used it: e^iπ + 1 = 0

Never mind, what about this? The prime density of the range 1 to x is x/lnx. WHAT?!?! Don’t worry. The prime density is the fraction of the number of primes from 1 to x divided by x. ln(x) is the natural logarithm, or log base e. So lne = 1. The prime density, represented as π(x), is approximately x/ln(x). What!!

Yes, that is a LOT of stuff, but you could just search it up online. But NEVER EVER TRUST WIKIPEDIA. How foolish I was to make an account!

Anyway, the next one o=is about how π is WRONG. See ya!

Eeeeeeeeek!

What is down the line of beauty, just underneath φ? It is, of course, e! By e! I don’t mean e factorial, but just e. e is a number of some confusing sort. e can be represented as:

e=lim x->infinity (1+1/x)^x

AAAAAAAAAAAAAAAAAAAAAAAAAAAA! Not to worry, it’s not as disgusting as it looks. Imagine it this way: you borrowed $100 from the author, and you have it for 1 year, and give it at the end of the year with 100% interest. You will give me 100(1+1) = $200 back.

Now imagine you paid quarterly for the year, with 25% interest. In the first quarter, that is $125, Halfway is $156.25, and at the end of the year, you have to pay me a tiny bit more that $244.14!!!!!

Tenthly (every 1.2 months) payment is a tiny bit more than $259.37. Now this is the crazy bit. You pay me every DAY, and I end up with more than $271.45. Phew! What if you paid me ALL THE TIME NON-STOP?!?! Then you will give me a little more than $271.82, or $100e!

That will give me $171.82 more than what I had before. Yeees, I’M RICH!!!!!! I AM SO TOTALLY THE RICHEST PERSON IN THE WORLD!!!!!!

Says you.

Money critic

Yes, THANK YOU! e isn’t just used in financial maths, but also in other stuff. Stick around …

I mean, where is this beautiful number even USED?!?!

Good question.

Good question.

The author

Yes, yes, thank you. Now, if you can remember ANYTHING from the post “Nothing beats the beauty of this!”, then you will know that φ is used in a lot of places and artwork. φ might be a drastic little number, but φ is very useful. For example, the Parthenon, which the top has crumbled a bit, if the top didn’t crumble, the rectangle of base equal to the base and height equal to the perpendicular height from the crumbled corner to the bottom, it’s coincidentally a Golden Rectangle.

Mona Lisa, the famous painting painted by Leonardo Da Vinci, has the Golden Ratio (aka φ) hidden inside the painting. It is believed that Da Vinci used the Golden Ratio to incorporate balance, harmony and beauty. What?

Flowers also use φ to decide how their petals grow. Also, shells like the Nautilus shell are based off φ.

Yes, φ is used everywhere. Can the human body POSSIBLY use φ???

This will be the final post about φ. I think I’ll write about e now. Not now, but NEXT TIME.

Next time! Next time! Don’t forget!

The White Witch, The Chronicles of Narnia: The Lion, The Witch, and the Wardrobe

Nothing beats the beauty of this!

What is the most beautiful number? It is, of course, phi. Also known as the golden ratio, it is represented by the Greek symbol φ , and is the most irrational of irrational numbers. φ can also be written as this:

(1+sqrt5)/2. This may look ugly, but it is it. φ is used everywhere, in buildings and artworks and even the Milky Way galaxy relates to it. φ isn’t just about buildings and artwork, but it is related to the Fibonacci sequence.

Here is a question: what starts with 1 1 2 3 5 8 13 21 34 55 89? It is, of course, the Fibonacci sequence. The spiral is also interesting, where you put two 1×1 squares together and put a square with dimensions 2×2 then 3×3 then 5×5 …

φ can also be represented as this ugly little thing:

φ = 1+1/ φ = 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( 1+1/( … and you’d never be able to get to the end!

This was Part 1. Get ready for Part 2. I GuaRanTEe that it will be awesome and awesome! SeE you.