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.

One is?

One!

Anybody

I mean, yeah. But in a prime/composite definition? It’s neither, the one exception, the golden, whatever.

Why is it not prime? Let’s have a look.

Take 25321, a prime number. 1 is also prime, right? So the factorization of 25321 would be 25321 and 25321*1 at the same time. But we have broken two golden rules:

  1. Every number, even prime numbers, have exactly one prime factorization
  2. A number cannot be both prime and composite at the same time.

Also, a proof that came in my head while writing:

You know Euclid’s Theorem that states ‘there is no prime because of this, that and stuff’? Full Proof:

Say the list of primes were finite, like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and that’s it. Multiply them all together, add one, it’s 6469693231 (2*3*5*7*11*13*17*19*23*29+1). Since this number is bigger than the “last” prime, it must be composite (some cases are when it actually is prime). But if you divide it by any “Prime” number, it will result in a remainder 1. So there must be some other “prime” out there that is prime.

Curtains, lights, action!

Say that one was prime, then Euclid’s Theorem to primes to 30 multiplication would still be 6469693231. But since 1 can divide 6469693231 evenly, and one is “prime”, Euclid’s Theorem has this weird exception, and that contradicts the fact that there are infinitely many primes.

1 isn’t composite either, since it doesn’t have any prime numbers that can divide it evenly.

So there you have it.

1 is not prime. So it’s not composite! Yay!

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)

Okay, WHAT?!?!

So, um, this is going to be ALL ABOUT FOREIGN LANGUAGES.

Vaay Bu eğlenceli olacak!
(Turkish for Whoo! This will be fun!)

No, this page ain’t going to be in a foreign language. But you can translate it.

This is about math stuff that uses other languages. Many are Greek, so watch out!

  • Delta (Δ), represents ‘the change in’
  • Zeta (ζ), which is the symbol for the Zeta Function
  • Theta (θ), represents angle
  • Pi (π, and NOT 🥧), which is the TOTALLY important circle constant
  • Sigma (Σ), the function used to add up terms
  • Tau (τ), the MORE IMPORTANT circle constant
  • Phi (φ), the Golden Ratio
  • Psi (ψ), the symbol for a wave function
  • Omega (ω), the first uncountable infinity
  • ∂, a partial derivative thingy
  • ∃, there exists
  • ∀, for all

I forgot to mention Hebrew letters. Here they are:

  • ℵ, symbol for infinite cardinality
  • ℶ, another symbol for infinite cardinality
  • WAIT, THAT’S

IT?!?!?!?!

Well, that’s it.. May your life be full of onnellisuus (happiness in Finnish).

Transcendental numbers!

What are those?!?!

Well, here’s your answer!

ME, THE AUTHOR

Transcendental numbers are numbers that you can’t do anything to it to get an integer. Besides power of 0, divide by itself, blah blah, blah.

More formal: In mathematics, a transcendental number is a real number or complex number that is not an algebraic number—that is, not a root (i.e., solution) of a nonzero polynomial equation with integer coefficients.

Look, I copied that all from Wikipedia, which I should not trust. π, e, Chaitin’s constant, and a WHOLE LOT OTHERS ARE TRANSCENDENTAL. Not √2, because if squared, it is 2.

φ also isn’t a transcendental number, since (2φ-1)^2 = 5. Wow!

I’ll be back with other stuff.

BYE!!!!!!!!!!!!!!!!!

😉😐😅😐😅😅😆😊😊😊😇😈😁😂😊😀😆😅😂😃😄😅😆😇😇😈😉😉😉😉😉😉😉😉😉😉😉😉😊😀😊😀😊😀😀😁😂😉😂😁😂😂😂😃😃😃😄😃😄😃😅😄😆😅😂😃😂😂😁😁😀😀😊😇😈😇😈😇😈😇😈😉😉😁😀😁😀😈😀😁😁😁

P=NP is a part of … Millennia!

How many Math questions are there? Obviously, there’s infinitely many. But how many problems can be solved quickly? A LOT! What about having the answer verified? MILLIONS OF BILLIONS! The P=NP question ask whether a question that can be verified quickly (P) can also be solved quickly (NP). Can you solve this?

Hello. The Millennium Questions are seven questions that gets you $1000000 if you solve it. There used to be 7, but Grigori Perelman already solved the Poincaré Conjecture. He denied EVERY PRIZE!!!!! CrAZy!

The other six want people to solve them. They are the Hodge Conjecture, Riemann Hypothesis, Yang-Mills existence and mass gap, Navier-Stokes existence and smoothness, and Birch and Swinnerton-Dyer conjecture. Have a go at any of them!

Look, I can’t explain everything, but the Clay Institute people are crazy. They went online, made a web about themselves, and then BAM! They’ve done it. I mean, I’m trying to solve the Riemann Hypothesis, but tough luck. NO CHANCE!

If you found anything for any of the questions, let me know on the Comments page. Details include:

  • Yer name
  • Which question
  • Answer
  • Ending (obviously)

GOOD LUCK!

Dear Archimedes,

I wish you DIDN’T EXIST!!! Well, that’s an overreaction, but please, how did you discover π? π is WRONG!!!!! Here’s proof. Listen CAREFULLY…

π is defined as the circumference of a circle divided by the diameter. I mean, WHO DRAWS A CIRCLE WITHOUT A RADIUS?!?!?!?! Obviously you use a diameter. Wow. But how?

Also, π radians is HALF A CIRCLE!!!!! 2π radians is a full circle, but it’s real frustrating to type or write that EVERY TIME!!!!!!!! Why couldn’t you have invented a number that equalled 2π?!?!

Luckily, I’ve got a few people to help me out. There is Vi Hart, Michael Hartl, Bob Palais, and a few other people. I will just replace π with τ. τ is 2π, and much easier to use in a lot of things. A LOT!

τ is the real circle constant, and is a whole circle. That’s a whole pie! τ is approximately 6.28, which is why Tau day is June 28. Not Pi day. TAU DAY!!! By the way, if you’re reading this on March 14, go have HALF A PIE.

Sincerely,

Gwang Ho KIM