Representation of Integers as Sum of Squares

Sum of two squares: 

• No integer of the form 4k + 3 is the sum of two squares.
• If each m and n are sum of two squares, then there product mn is also a sum of two squares.
• Thuw's lemma: Let p be a prime and a an integer, which is coprime to p. Then the linear congruence ax ≡ y (mod p) has the solution (x_0, y₀) such that                                              0 < |x₀| < √p and 0 < |y₀| < √p.
• Fermat's lemma: An odd prime p can be represented as a sum of two squares iff p ≡ 1 (mod 4).
• A positive integer n > 1 can be represented as sum of two squares iff either n has no prime factor congruent to 3 (mod 4) or if it has a prime factor congruent to 3 (mod 4) then it occurs to an even power in the prime factorization of n.
• Every odd prime is the difference of two squares in one and only one way.

Sum of three or more squares: 

• Any integer of the form 4ⁿ (8m + 7) integers m, n ≥ 0 is not a sum of three squares.
• If p is a prime number. Then there exist integers a, b, c atleast of which are non-zero such that  a² + b² + c² ≡ 0 (mod p)
• Euler's lemma: If each of two positive integers m and n is a sum of four squares, then their product mn is also a sum of four squares.
• Any prime number p can be written as sum of four non-negative squares.
• Legranges theorem: Every integer > 1 can be represented as the sum of four non-negative squares.
• Aubry theorem: There are infinitely m any primes each of which is a sum of three distinct squares

Arithmetical function: A real or complex valued function defined on the positive integers is called an arithmetical function.

Mobius Function:
The Mobius function μ is defined as
                                    μ(1) = 1
     If  n > 1, write n = p₁^a1 … p_k^ak , then 
                                    μ(n) = (-1)^k if a₁ = a₂ = … = a_k = 1
                                    μ(n) = 0, otherwise.

• μ(n) = 0 iff n has a square factor > 1.
• If n ≥ 1, we have

                                   ∑μ(d) = [1/n]