Factoring large numbers into primes | Famous Math Problems 1 | NJ Wildberger

This is the first video of a new series, which will discuss a wide variety of famous (and perhaps not so famous) mathematical problems, ranging from antiquity to modern times. I am N J Wildberger, a pure mathematician at UNSW, the discoverer of Rational Trigonometry and Universal Geometry, and a big fan of YouTube, the future of free on-line education!

After introducing the series, we begin with FMP #1: how to factor a particular number z (10 to the 10 to the 10...altogether ten times, plus 23) into primes. We remind viewers about primes and the Fundamental theorem of Arithmetic, and have a look at explicit factorizations related to our basic problem.

Then we introduce briefly modular arithmetic and the theorems of Fermat and Euler. This is used to show that 7 does not divide z. Some comments on difficulties with arithmetic with big numbers follows, and then some exercises give some challenges for the viewer.

Thanks to Professor Saad El-Zanati from Illinois State University for suggesting this series. Thanks also to Steve Sexton from Exchange for giving permission to use their music track Highlights.

Video Chapters
00:00 Introduction
2:37 Difficulty scale of Problems
9:46 Find the prime factorization of z
14:30 Easier factoratizations 10^n+23
18:21 Wolfram Alpha
22:41 Modular arithmetic (Gauss)
29:00 Does 7 divide z ?
34:02 Some division Exercises

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Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

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[njwildberger]

That's a quite valid point. So I should say something like: the ``numbers less than z'' that we usually assume exist have in fact a very problematic existence. Or I could say: while it is possible to discuss small numbers, say less than 10^10^10, and it is possible to discuss some large numbers, such as z, it is not the case that we have a consistent theory of numbers between these.
It is rather onerous and delicate to talk accurately about this issue, which perhaps explains why we seldom do

[Pygmygerbil]

It is a blessing to have a professor like nj wildberger in your life

[tylershepard]

As a computer science major, I absolutely love your videos. Makes me think about these problems.

[acudoc]

What a gem of a teacher! Confirms my belief that teachers are born, not made...

[njwildberger]

Physical existence is relevant to mathematics, for the simple reason that we want to write things down. Whether it be on a piece of paper or a hard drive, our thoughts are made concrete once we formulate them clearly using a language we have agreed on. It follows that when we get to the limits of what we can write down, we are also getting to the limits of what we can clearly call mathematics.

Thoughts which are disconnected from symbols ought to be viewed suspiciously in mathematics.

[MichaelHejazi]

I am very grateful for your work N J Wildberger!

[robharwood]

Rewatched this. *Dr. Wilberger*, this remains a really great video: As an intro to your series, as an intro to this math problem, as a pedagogical video. Added it to my favourites. :-)

[venkateshwarlubommagani]

Dear Sir, I am not a professional mathematician, but an Engineer, who has a bit of needed understanding of mathematics for Engineering courses.

Your videos have been very insightful and have immense pedagogical value. thank you for making them available for general public.

I would like to bring to your kind consideration, the below line of argument, in the context of concluding remarks at the end of the above video "Factoring large numbers into primes | Famous Math Problems 1 | NJ Wildberger". it is a bit perplexing to see the conclusion that "the inability of a number to fit in the UNIVERSE, when written in some form, implies that the number doesn't exist". To be frank even this statement of your is also right, yes these numbers does not exist in physical sense, in written form. But, the number exists in conceptual form, satisfying basic axioms and rules of logic (for ex. every natural number will have a successor; every natural number is either even or odd etc. these rules do not demand physical existence as necessary pre-requisite). the best example is complex number: imaginary numbers do not instantiate COUNT of any physical quantity.

But the CONCEPT of number, or a theorem, does not in any way derive its truth value from being able to be written in a given space (whether the space is a paper or a paper of size of the UNIVERSE); I am sure you very well know, that the numbers, THEOREMS in mathematics are concepts, relations; to be precise "mathematics is a body of UNDISPROVED concepts and ideas that are consistent with each other", I think they don't derive their TRUTH VALUE from being able to be written on PAPER, ON A BOARD, IN FOOT BALL GROUND, ON EARTH, ON SOLAR SYSTEM, ON GALAXY or EVEN IN THE UNIVERSE.

the statement that "The prime factorization (of the big numbers) does not exist", and "The fundamental theorem of arithmetic's is not true" may be too hasty conclusions.

lets assume that we have a number with factors, which does not fit on a paper, that inability to fi in a paper (we can get whole bunch of them), that in any way does not stop that number to have PRIME FACTORS. As "THE PROPERTY OF HAVING FACTORS" does not depend on the number's ability to fit on PAPER, like wise, it does not depend on its ability to fit in the paper of the size as big as "Foot ball stadium", or its ability to fit in a paper AS BIG AS UNIVERSE.

As long as the AXIOMS and the principle of the LOGIC holds true, the size of the number should not matter.

this is written with my little understanding, it would be an honour to hear your opinion on this line of argument. .

you are a ocean of knowledge of mathematics, I am just a drop asking the ocean a question, expecting some enlightenment with your answer.

grateful to have your lectures, thank you sir !!!

- Venkateshwarlu Bommagani

So here what I noticed about patterns :
It is always divisible by 3 because 10^x+23 = 1+2+3=6 which is divisible by 3
if you divide (10^x+23)/3 you will get 3*(x-2)41 where 3*(x-2) is the number of repetition of 3, for example ((10^5)+23)/3= 33341 with that expression is 3*(5-2)41=33341
11 factor repeated in odd powers,
41 repeated in 5k+2
13 repeated in 6k+4

[sanchomathew]

Great work sir... These are the type of videos.. which makes people really understands the beauty of mathematics....

[njwildberger]

There is much to be gained by thinking scientifically, especially if the alternative is thinking philosophically. The former motivates us to observe, make calculations and abstain from wishful feel-good thinking, the latter is best suited for ethical discussions and cocktail conversations.

[njwildberger]

No, I do allow that possibility. You are allowed to use whatever tools you like. So when I say D=3 it means you can expect to take (very roughly!) about a week to figure it out, using all tools you can get your hands on.

[viamathesis]

Indeed. Taking a detour into the realms of metaphysics will be necessary to clarify what is meant by "existence". What does it mean to exist? What is being? This of course opens a whole new area of problems, but by thinking about them we can clarify our ontology in mathematics.

[atulkumthekar]

excellent explanation. I was reading many videos, lectures and links for Shors algorithm for quantum computing and needed to understand factorization as a pre-requisite. This was the best.

[njwildberger]

The scientific approach is to only adopt what is necessary to understand what we see. There is no reason to invoke that are completely and forever invisible to us. In mathematics, that ensures that we stay closer to the true nature of things. Your mystical objects, my mystical objects--- who needs them??

[njwildberger]

That is right. As I will discuss at some point in the MathFoundations series, numbers have, wrt a given arithmetical system, a complexity/information content. Bigger numbers have greater complexity in general, but the relation is highly sporadic. Numbers like z form islands of simplicity (low complexity) in vast seas of more complex numbers. Eventually, as we keep going with larger numbers, everything just dissolves into a haze of complexity.

In particular, ``infinite sets'' are a joke.

[BillyBarret]

Norman i have seen nearly all of your vid series and i am very pleased that you chose to take the time to do this series, also i am eager to see the next in the math foundations series which i must say you have done the mathematics community a great service for your work in explaing these areas in mathematics. keep up the great work and thank you very much. PS: my focus for two years has been in the d=8 and d=9 categories, and cant wait to hear what you have to say about FLT, its my main focus.

[njwildberger]

I want our theorems to be actually true, not just theoretically true if such and such a philosophical position holds. I am interested in a mathematics which everyone can agree on, independent of their religious or philosophical position, and propose to investigate only this.

Let Philosophy expand and incorporate all those interesting musings about ``infinities'' and ``things which cannot be spoken about or written down''.

[njwildberger]

Actually they are meant to be viewed in order. However you can browse through the list of titles if you pull up the Playlist MathFoundations, to see what might interest you if you want to dabble. As for philosophy, I admit that my aim is to try to minimize that in mathematics!

[njwildberger]

An excellent post, thanks very much. I like your nice notation which captures the situation very well. Do you have proofs of any of these claims? Can anyone extend this fine list?