Parity (mathematics) - Higher Mathematics

Higher Mathematics

The even numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.

All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.

The squares of all even numbers are even, and the squares of all odd numbers are odd. Since an even number can be expressed as 2x, (2x)2 = 4x2 which is even. Since an odd number can be expressed as 2x + 1, (2x + 1)2 = 4x2 + 4x + 1. 4x2 and 4x are even, which means that 4x2 + 4x + 1 is odd (since even + odd = odd).

Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 1014, but still no general proof has been found.

The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.