Theorem of Schur

Schur's theorem states that if a positive integer \( n \) can be expressed as the sum of two squares in two different ways, then any prime number dividing both of those representations must be congruent to 3 modulo 4. In simpler terms, it links the ways a number can be broken down into two squares with properties of its prime factors. Specifically, it restricts how primes dividing numbers that are sums of squares must behave, showing a deep relationship between prime numbers and the way integers can be represented as sums of squares.