2n

2n denotes two times the integer n. When n ranges over the integers, the values 2n are exactly the even integers: ..., -4, -2, 0, 2, 4, 6, .... If n ranges over the natural numbers (often including zero), 2n yields the nonnegative even integers: 0, 2, 4, 6, ...

A key property is that 2n is always divisible by 2. Consequently, an integer m is even if and only if there exists an integer n such that m = 2n. The relationship also provides a simple inverse for even numbers: n = m/2, which is integral precisely when m is even. In modular arithmetic, 2n ≡ 0 mod 2 for all integers n.

The map n ↦ 2n is a basic way to generate the even integers and is used across number theory and algebra. For arithmetic operations, gcd(2n, 2m) = 2 gcd(n, m), and divisibility properties extend to multiples of two. In sequences and counting, 2n commonly appears in formulas involving pairs or double quantities.

In brief, 2n is a simple linear expression that encodes the set of even integers through integer inputs, with a direct interpretation in terms of parity, divisibility, and integer factorization.