426 Digital sum and digital root
15.3 The digital roots of the powers of 2 ...........
n 0123 4 5 6
2 n 1248163264
D(2n)1248 7 5 1
It follows that the sequence of digital roots of 2 nis periodic with
period 6.
sequence of digital roots
2 n 1 , 2 , 4 , 8 , 7 , 5
3 n 1 , 3 , 9
4 n 1 , 4 , 7
5 n 1 , 5 , 7 , 8 , 4 , 2
6 n 1 , 6 , 9
7 n 1 , 7 , 4
8 n 1 , 8
9 n 1 , 9
The sequence of the digital roots of 2 n+1is also periodic with period
6:
2 , 3 , 5 , 9 , 8 , 6 , ....
Digital roots of Fermat numbers
To find the digital root of the Fermat numberFk=2^2
k
+1, we need only
find 2 kmod 6. Now, it is clear that 2 k≡ 1 or2mod3according askis
even or odd. Therefore, 2 k≡ 4 ormod6according askis even or odd.
From this, we have
D(Fk)=
{
8ifk≡0mod2,
5ifk≡1mod2.
The digital roots of the Fibonacci numbers form a sequence of period
24:
1 , 1 , 2 , 3 , 5 , 8 , 4 , 3 , 7 , 1 , 8 , 9 , 8 , 8 , 7 , 6 , 4 , 1 , 5 , 6 , 2 , 8 , 1 , 9 , 1 , 1 ,....
