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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 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.


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 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 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.

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