doubly even number∗
1and2and4†
2013-03-22 1:13:38
A doubly even number is an even number divisible by 4 and sometimes greater
powers of two. If n is a doubly even number, it satisfies the congruence n ≡ 0
mod 4. The first few positive doubly even numbers are 4, 8, 12, 16, 20, 24, 28,
32, 36, 40, listed in A008586 of Sloane’s OEIS.
In the binary representation of a positive doubly even number, the two least
significant bits are always both 0. Thus it takes at least a 2-bit right shift to
change the parity of a doubly even number to odd. These properties obviously
also hold true when representing negative numbers in binary by prefixing the
absolute value with a minus sign. As it turns out, all this also holds true in
two’s complement. Independently of binary representation, we can say that the
p-adic valuation of a doubly even number n with p = 2 is always 14 or less.
All doubly even numbers are composite. In representing a doubly even
number n as
π(n)
Y
pi ai ,
i=1
with pi being the ith prime number, a1 > 1, all other other ai may have any
nonnegative integer value.
If n is doubly even, then the value of τ (n) (the divisor function) is even
except when all the nonzero ai in the factorization are greater than 1.
Whereas (−1)n = 1 whether n is singly or doubly even, with the imaginary
unit i it is the case that in = 1 only when n is doubly even.
∗ hDoublyEvenNumberi
created: h2013-03-2i by: h1and2and4i version: h40718i Privacy
setting: h1i hDefinitioni h11A63i h11A51i
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You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
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