Fermat's Theorems

Fermat's Last Theorem

The Diophantine equation aⁿ + bⁿ = cⁿ has no solutions when n > 2.

This is easy to prove for various specific values of n (proof when n=3; when n=4), and has been proved recently for all n (but that was a very difficult proof that required the use of a computer).

Fermat's Little Theorem

Fermat's "Little" Theorem is sometimes abbreviated FLT, which is confusing because the same abbreviation might mean his "Last" theorem.  This "little" theorem is a special case of Euler's Totient Theorem, and the proofs are almost identical.

a^p-1 = 1 (mod p), where p is prime and a≠0 (mod p)

Proof: take a to be any positive integer coprime to p.
Consider the set {a, 2a, 3a, ..., (p-1)a}.
No two elements of this set are congruent mod p (see below*), so their residues mod p must be {1, 2, 3, ..., p-1} in some order.
By equating the product of the numbers in the first set with the product of those in the second set, mod p,
a^p-1(p-1)! = (p-1)! (mod p)
Since (p-1)! is coprime to p, we can divide both sides by (p-1)!, mod p, proving Fermat's Little Theorem.

* Why are no two elements of {a, 2a, 3a, ..., (p-1)a} congruent, mod p?
Suppose a is a number coprime to p, and b and c are two different numbers in the range [1,p-1] and ab=ac, mod p.
Then ab-ac=pk, where k is an integer.
a(b-c)=pk, and since b-c is smaller than p in absolute value, a must be larger than k in absolute value.
a and p have no factors in common, so a|k, but a can't divide k, since a is larger than k, a contradiction.

Other "Fermat" Theorems

An odd prime number is a sum of two squares if and only if it is congruent to 1 modulo 4.  This is given amongst a bunch of theorems and definitions about Ring Theory, and explained in great detail in the Sum of Four Squares page.

Related Pages in this website

Euler's criterion for determining whether a number is a quadratic residue (mod p), or in other words, if a number is the square of some other number (mod p).

Proof of Fermat's Last Theorem for n=3, attributed to Euler

Proof that there is no solution to the Diophantine equation A⁴+B⁴=C², which means Fermat's last theorem is true for n=4. 

Squares.

Euler's Totient Theorem -- a generalization of Fermat's Little Theorem

The proof that primes of the form 4k+3 can be expressed as the sum of four squares uses Fermat's Little Theorem.

Group - a set closed under one operation.

A bunch of Ring Theory theorems and definitions.

The webmaster and author of the Math Help site is Graeme McRae.
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