Perfect Squares proves the
following theorems:
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.
Sum of Squares proves the following
very interesting factoids:
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 xp-1 - 1 = 0 (mod p), where p is an odd prime
Fermat's Little Theorem: ap-1 = 1 (mod p), where p is prime and a¹0
(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.
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).
Irrationality Proofs --
Proofs that p 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 a2, b2, c2 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.
