Theorem 0: If N>1 is not a perfect square, then sqrt(N) is irrational -- i.e.
sqrt(N) cannot be expressed as a/b, where a and b are integers -- and two
corollaries: If sqrt(N) is rational, then N is a square, and any integer which
is a ratio of squares is a square.
Theorem 1: If a is a non-zero integer, and b is an integer, and a is a
perfect square, then the following statement is true: ab is a perfect square if
and only if b is a perfect square.
Theorem 2: If a and b are relatively prime, then ab is a perfect square if
and only if both a and b are perfect squares.
Every positive integer is the sum of four squares. This proof uses the
next two factoids:
Primes of the form 4k+1 can be expressed as the sum of two squares
Primes of the form 4k+3 can be expressed as the sum of four squares
Procedure to find a number, a, such that a^2 = -1 (mod p), where p is of the
form 4k+1
Polynomial Roots x^{p-1} - 1 = 0 (mod p), where p is an odd prime
Fermat's Little Theorem: a^{p-1} = 1 (mod p), where p is prime and
a≠0 (mod p)
Proof that there is at least one solution to y^2=-1-x^2 (mod p)
The Complex Product Identity, which show that the product of two numbers,
each of which can be expressed as the sum of two squares, can itself be
expressed as the sum of two squares.
The Quaternion Identity, which shows the same thing, but with four squares
instead of two.
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).
Irrationality Proofs -- Proofs
that π and e are irrational.
Pythagorean Theorem
Prove that the area of a right
triangle with integer sides is not a perfect square.
Arithmetic Sequence of Perfect
Squares, page 3 -- If a^{2}, b^{2}, c^{2} are in
arithmetic sequence, why is their constant difference a multiple of 24?
Look at the second answer to this question for the relationship between
Pythagorean Triples and this arithmetic sequence of squares.
Squares and Fourths -- can
the sum of two consecutive fourth powers equal the sum of two consecutive
squares?
Fermat's Therems.