
Fermat's Little Theorem (2013) [video] - ColinWright
https://www.youtube.com/watch?v=XPMzosLWGHo&feature=youtu.be
======
mixedmath
There is an elementary number theory textbook by the famed number theorist and
Ramanujan scholar Dr. George Andrews called Number Thoery,
[http://www.amazon.com/Number-Theory-Dover-Books-
Mathematics/...](http://www.amazon.com/Number-Theory-Dover-Books-
Mathematics/dp/0486682528)

It contains proofs of very many results from elementary number theory, like
Fermat's Little Theorem here. The reason this book is different from any other
is because the proofs are all combinatorial, like counting beaded necklaces or
otherwise. It is also a Dover book, meaning it costs very little. It is also
one of the books I used as reference to create the elementary number theory
course that I sometimes teach.

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clear-as-mud
This is why I despise mathematicians. Everything was going smoothly, until
they obscured the whole concept behind that awful notation. Why? Notation is
the destroyer of understanding.

Letter, pipe, letter, superscript. Yeah, real nice. Clear as mud.

~~~
ColinWright
If you choose not to invest the time learning the notation, you will forever
be limited to writing everything out in longhand. Try reading maths from the
1600s or 1700s and you will come to appreciate the value of notation.

And don't start down the road of thinking everything can be made unambiguous.
It is context dependent, and always will be. Look at computer code, look at
something deep in the bowels, and tell me what the "+" sign means. Is it
integer? Float? List? Or has it been over-ridden by some library you've loaded
somewhere else? Even finding where it's been defined can be a challenge. You
have to trust that the author has played fair, and that your intuitive
understanding that "+" is doing something consistent with the concept of "+"
in numbers is close enough.

You could write out all the math notation in longhand, but it won't aid your
understanding. If you actually follow the video then you will follow the
notation. But there's a reason why we say "read like math" \- it takes time
and effort to gain understanding. It's not a novel, it's not a comic.

    
    
        The only way to learn mathematics
            is to do mathematics.
                -- Paul Halmos
    
        "There is no Royal Road to geometry,"
            -- Euclid, in reply to King Ptolemy's
               request for an easier way of learning
               mathematics.

~~~
jonsen
Well said and all, but notation _is_ a destroyer of understanding, _if_ you
are not well versed in reading math. There is a very abrupt jump in the video
from the pleasant down to earth demonstration using picture language when the
"cold" math notation is just thrown at you in the end.

~~~
ColinWright
If you are not well versed in reading math then you have two options. One is
to forever remain fuzzy and imprecise in your understanding, the other is to
learn the notation you need, see how it matches your internal understanding,
and then use it as an enabler.

If the notation destroys your understanding, then you didn't understand it in
the first place, you only had a vague sense of satisfaction with a pseudo-
understanding. This is akin to thinking you understand an algorithm, but not
being able to code it.

I have every sympathy with people who find the notation a barrier, because so
did I. Over the years, indeed, the decades, I have come to appreciate notation
as a way of concisely expressing deep and complex ideas. Try not using
notation - your brain quickly turns to mush as you try to follow hundreds of
words that could otherwise be expressed in a mere dozen symbols.

Archimedes in "Measurement of the circle" wrote:

    
    
        The area of any circle is equal to a
        right-angled triangle in which one of
        the sides about the right angle is equal
        to the radius, and the other to the
        circumference, of the circle.
    

Today we would write:

    
    
        Area = (1/2)⋅r⋅(2πr)= π⋅r^2
    

People have a phobia about notation, but the solution isn't to avoid notation,
it's to show that it is powerful, and that overcoming the phobia is one way of
gaining access to that power.

 _Edited to change the example from Newton 's Principia to Archimedes_

~~~
jonsen
My point was, that the way notation is just thrown at you in the video can
only add to the phobia for the uninitiated. The video is clearly aiming at
basic pedagogic explanation, and it is all destroyed in the end. The math
versed who easily read that math without explanation do not need the bead
juggling. The pedagogy fails in the end.

~~~
ColinWright
The explanation of the theorem has got as far as it can go by that point.
There's a choice to be made. Avoiding notation means that those who could now
connect the explanation with things they have already seen is lost. Avoiding
notation also means that it's just all been a show, and there's no real take
away except that, well, this guy played with beads an counted things.

If someone is truly notation-phobic then there is nothing to be done.

Having the notation means that those in the middle ground have a chance to see
real maths in action, and may be inspired to do more. The genuinely notation-
phobic will never advance in math. For those who know the theorem this is a
cute visualisation. People will only learn when they are ready, and all
pedagogy needs to be targeted at those who are ready for it.

I suspect we agree more than people reading this might think, but we might
disagreed over the perceived purpose of the video. I see it as getting people
engaged with the thought processes, and then showing that it's really math,
and look, here are the formulas that come out of it. It's an opportunity to
learn about the formulas. For the genuinely notation-phobic, there is no real
hope, except for them to see that this is a cool thing their phobia is
preventing them from understanding better, so maybe getting over the phobia
would be worthwhile.

I could do a complete case analysis, but this isn't the place, and I don't
have the time. Would you suggest not having the notation at all? What would
then be the point of the video? Who would gain?

~~~
jonsen
Yes, we agree a long way. I'm sure the video works well in its intended
greater context, as you describe here. My points are about taking the video at
face value out of context, which is probably what lead to clear-as-muds
comment.

~~~
britcruise
Hey there, I created this video. It was just one part of a longer lesson I was
trying on "Random Algorithms" because it involved Fermat's primality test...so
in this case we really needed that formula to try it out which is why I added
it at the end.

[https://www.khanacademy.org/computing/computer-
science/crypt...](https://www.khanacademy.org/computing/computer-
science/cryptography/random-algorithms-probability/v/randomized-algorithms-
prime-adventure-part-8)

------
CurtMonash
I love Fermat's Little Theorem, both for its own sake and because it's an
intermediate step in proving the Two Squares Theorem. My equivalent to
"counting sheep" used to be to review the proof of the TST.

The simplest proof I know of Fermat's Little Theorem is induction, assuming
that we already know the Binomial Theorem. Suppose a^p == a. (== is my symbol
for "is congruent to mod p" in this post.) Expand (a+1)^p, and note that every
binomial coefficient except the first and last has p in the numerator but not
the denominator. Hence (a+1)^p == a^p+1 == a+1.

Cleaning up the proof from there is trivial.

~~~
baggers
This question is as someone who is mathematically curious but not yet adept.

In programming we slowly gain a big grab-bag of patterns and approaches to
certain problems and build an intuition of what to apply where. It doesn't
nearly cover your full experience but helps break problems down.

Do you find there is an analog to this with theorems? If so what's the
essentials from your 'grab bag' and, beyond just reading more, what practices
help build your feeling of where to use them?

~~~
ColinWright
Yes, there is, but I've never thought of it in those terms, and I can't
enumerate them easily. There are techniques to apply, approaches to try, and
connections to make that can help. But I don't know any way of building
intuition other than by actual doing.

Tim Gowers writes well about how to build mathematical knowledge, techniques,
and a library of tools.

~~~
baggers
Thanks, that's good to know.

------
dkarapetyan
Reminds me of Burnside's Lemma
([https://en.wikipedia.org/wiki/Burnside%27s_lemma](https://en.wikipedia.org/wiki/Burnside%27s_lemma)).

~~~
JadeNB
Indeed, the idea of the proof of Burnside's lemma is essentially the same as
what is often called the necklace-counting proof of Fermat's little theorem:
[https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_th...](https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem#Proof_by_counting_necklaces)
.

~~~
ColinWright
Yes, and the submitted video is that necklace counting proof, hence the video
reminding them of it.

~~~
JadeNB
Heh, sorry about that. I was on a low bandwidth connection, and couldn't watch
the video.

------
S4M
Am I the only one who doesn't get it? My background is in maths and I am (was)
able to prove formally Fermat's little theorem, but I got lost around minute
3:08, and am not clear why the fact that 5 is prime means that any combination
must take 5 rotations to return to itself. I can see that it is true but the
reason is not clear to me.

~~~
gjm11
Suppose the smallest (positive) number of rotations it takes to return is d.
Then the numbers of rotations that have that effect are 0, d, 2d, 3d, etc. But
clearly 5 rotations take you back where you started, so 5 is one of those
numbers; that is, 5 is a multiple of d. Since 5 is prime this means either d=1
or d=5.

My second sentence above is a bit of a handwave. If you want to get formal
about it, here's one way: suppose d isn't a multiple of 5; then note that
there's an integer a such that ad = 1 (mod 5). Then ad rotations have the same
effect as 1 rotation; but ad rotations do nothing, so d=1.

Now the handwaving is concentrated in the "then note that ..."; if _that_
isn't sufficiently obvious, consider ad for a=0,1,2,3,4 and note that no two
can be equal mod 5 because if ad=bd mod 5 then (a-b)d is a multiple of 5, but
neither factor is a multiple of 5 and 5 is prime. So: five numbers, all of
them different mod 5, so we must have one each of each congruence class mod 5;
in particular, one of them is 1 mod 5.

~~~
S4M
Yes, that's a good explanation (I suspect your first paragraph is at the core
of Fermat's little theorem's demonstration), it's just a shame it was left out
of the video - but maybe because this part would require its own video.

------
mrcactu5
Fermat Little Theorem is one of my favorite results. Here is a short proof by
Lionel Levine it is only 2 pages
[http://www.math.cornell.edu/~levine/fermat.pdf](http://www.math.cornell.edu/~levine/fermat.pdf)

He also get formulas like: pq divides a^pq - a^p - a^q + a

~~~
JadeNB
> He also get formulas like: pq divides a^pq - a^p - a^q + a

Incidentally, assuming you mean p and q to be distinct primes, this latter
formula is almost just another instance of Fermat's little theorem: we have
that q divides (a^p - a)^q - (a^p - a) and p divides (a^q - a)^p - (a^q - a)
(that's the little theorem), and then that q divides (a^(pq) - a^q) - (a^p -
a)^q and p divides (a^(pq) - a^p) - (a^q - a)^p (that's essentially the
binomial theorem); so both p and q divide a^(pq) - a^p - a^q + a.

------
zurtri
There is an awesome book entitled "Fermat's Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem"

(you can google it for a link to a supplier)

Well worth the read. Tells the whole story of the famous "Fermat's last
Theorum" problem and is extremely well written.

~~~
curryhoward
You may know this, but for those who might not: Fermat's Last Theorem and
Fermat's Little Theorem are two totally different theorems (both about number
theory).

Fermat's Little Theorem: For any integer a and prime p, a^p is congruent to a
mod p.

Fermat's Last Theorem: For any integer n > 2, there are no integers a, b, and
c such that a^n + b^n = c^n.

The Little Theorem has been known for hundreds of years and is easy to prove.
The Last Theorem, on the other hand, was only finally proven in 1994 by Andrew
Wiles after countless failed attempts by mathematicians before him.

