twicesquares

Twice-squares are natural numbers of the form 2k^2, where k is an integer. The positive members begin 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, and so on. The value 0 is sometimes included as a trivial example when k = 0.

Equivalent characterizations: n is a twice-square iff n/2 is a perfect square. In prime factorization, if n = 2^e · m with m odd, then e is odd and all prime factors of m have even exponents; equivalently, m is a square. Thus n = 2 · s^2 for some s.

Congruence properties: If k is even, n ≡ 0 mod 8; if k is odd, n ≡ 2 mod 8.

Geometric and algebraic interpretation: n = a^2 + b^2 with a = b, so twice-squares are the norms of Gaussian integers a + ai. This links them to representations of numbers as sums of two squares and to related Diophantine considerations.

See also: sums of two squares, Gaussian integers, and representations of integers as sums of squares.