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The standard definition of "even number" can be used to directly prove that zero is even. A number is called "even" if it is an integer multiple of 2. As an example, the reason that 10 is even is that it equals 5 × 2.
The even numbers form an ideal in the ring of integers, [13] 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 ...
The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott, [15] which directly implies that every even number n ≥ 4 is the sum of at most 4 primes. [ 16 ] [ 17 ]
The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form 2 p−1 M p, where M p is a Mersenne prime. [1] The perfect number 6 comes from p = 2 in this way, as 2 2−1 M 2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added.
For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -8..." In logic, the extension of a predicate is the set of all things for which the predicate is true. [49]
A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics , coding theory (see even codes ), among others.
The numbers in the right column are the inversion numbers (sequence A034968 in the OEIS), which have the same parity as the permutation. In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd ...