Algodoo

Odd + Odd = Even: Adding two odd primes always results in an even number greater than 2, which is never prime (e.g., 3 + 5 = 8).

Odd + Even = Odd: Adding an odd prime and the even prime (2) results in an odd number, which can be prime (e.g., 2 + 3 = 5, 2 + 5 = 7).

Goldbach's Conjecture: Every even integer greater than 2 can be written as the sum of two prime numbers (e.g., 10 = 3+7, 10 = 5+5) – still unproven but widely believed.

Sum of Two Squares: A prime number \(p\) can be expressed as the sum of two squares (\(x^{2}+y^{2}\)) if and only if \(p=2\) or \(p\equiv 1\quad (\mod 4)\) (e.g., \(5=1^{2}+2^{2}\), \(13=2^{2}+3^{2}\)).