Help with "Fermat's Little Theorem?"?

Question

My Calc Assg. over break is to come back to school in January and be able to speak intelligently on "Fermat's Little Theorem," and there could be a quiz that's going to make us prove it. No one in my class has ever heard of this before, and it's not in the book.
I found a ton of stuff on it online, but it makes no sense and its really weird. How should I learn this??

Oh my goodness, I'm very confused.

Answer 1

The answerers above have confused Fermat's Last Theorem

with Fermat's Little Theorem. Here is how the Little Theorem

works: As an example let's choose a prime number like 7.

The remainders you get when you divide a number by 7

are 0,1,2,3,4,5,6. You'll notice that the remainders you get

when you divde 1^6,2^6,3^6,4^6,5^6,6^6 by 7 are:

1,1,1,1,1,1 all ones. This led to the theorem: If p is a

prime number, and p does not divide a, then p will divide

a^(p-1) - 1.

Answer 2

In a nutshell,

You are familiar with the Pythagorean Theorem:
a² + b² = c²

You can solve the Pythagorean Theorem with integers like
a = 3, b = 4, c = 5
or
a = 5, b = 12, c = 13
etc.

These are called Pythagorean triples.

Fermat wondered if you could do the same thing with cubes?
a^3 + b^3 = c^3

Or with higher powers:
a^4 + b^4 = c^4

He conjectured that you could *not* come up with integer solutions for higher powers. That you could only do it with powers up to 2.

Supposedly Fermat figured out a theorem to prove this conjectureand wrote a note in the margin of a book saying something to the effect of "I have proven this, but there isn't enough room to write it here". Shortly after that he died.

Today no one has come up with a proof of this conjecture, simple or otherwise.

Edit:
D'oh! I confused it with Fermat's Last Theorem... I just realized you asked about Fermat's Little Theorem. Nevermind then...

Try reading more here... Euler was the first to come up with a formal proof of Fermat's *Little* Theorem.
http://fermatslasttheorem.blogspot.com/2005/08/fermats-little-theorem.html

Answer 3

don't worry about trying to prove it. proffessional mathematicians have been trying for decades, with no success.
The joke is that Fermat himself only said "I have a marvelous proof that won't fit in this little margin" but he never told anyone what it was... which is why it's so plaguing for other mathematicians.

More to the point, be able to talk intelligently about why it can't be proved, or what problems have been encountered, and failed proofs of the past.

Answer 4

What a wonderfully exciting project you have
You cannot do better than starting at the site given earlier

Answer 5

http://en.wikipedia.org/wiki/Fermat%27s_little_theorem