Fermat's Little Theorem (2013) [video]
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Fermat's Little Theorem (2013) [video]
1–10 of 28 posts
Re: Fermat's Little Theorem (2013) [video]
#2Reminds me of Burnside's Lemma (https://en.wikipedia.org/wiki/Burnside%27s_lemma).
Re: Fermat's Little Theorem (2013) [video]
#3Fermat 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
He also get formulas like: pq divides a^pq - a^p - a^q + a
Re: Fermat's Little Theorem (2013) [video]
#4There 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.
Re: Fermat's Little Theorem (2013) [video]
#5There 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.
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.
Re: Fermat's Little Theorem (2013) [video]
#6Fermat 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 He also get formulas like: pq divides a^pq - a^p - a^q + a
> 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.
Re: Fermat's Little Theorem (2013) [video]
#7Reminds me of Burnside's Lemma ( https://en.wikipedia.org/wiki/Burnside%27s_lemma ).
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... .
Re: Fermat's Little Theorem (2013) [video]
#8I 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.
Re: Fermat's Little Theorem (2013) [video]
#9There 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.
My best friend from graduate school (which was the Harvard Math department at a time Wiles was an assistant professor) used to stay with me when he visited his employer in NYC after moving to Israel. He would take an overnight flight and arrive, quite blearly, early in the morning. The day of a visit, the NYT had the story. And so I greeted him at the door to my apartment, not with my usual warm hello, but rather the words "Wiles proved Fermat's Last Theorem".
Re: Fermat's Little Theorem (2013) [video]
#10There 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/...
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.