For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions).
The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed. Group theory Rubik's Revenge in solved state
Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10 18, but still no general proof has been found. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. All known perfect numbers are even it is unknown whether any odd perfect numbers exist. 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.Īll prime numbers are odd, with one exception: the prime number 2. 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. Then an element of R is even or odd if and only if its numerator is so in Z. − 2 ⋅ 2 = − 4 0 ⋅ 2 = 0 41 ⋅ 2 = 82 may be called odd.Īs an example, let R = Z (2) be the localization of Z at the prime ideal (2).
