Theorem on the Distribution of the Figures 0, 1 and
2 in the Various Orders of the base 3
Representations of kth Powers of Integers
Roberto Vacca - February 9, 2003
Let us consider a generic natural number n in its base 3 representation

n =   (h, n) 3h
, with  (h, n) = 0 or 1 or 2
0
The base 3 representation of the kth power of n, with k a positive integer, is

n =  k (h, n) 3h
k
, with k (h, n) = 0 or 1 or 2
0
We observe first that
k (h, n) = k (h, n +3h+1)
or, in other words, that k (h, n) is periodic with period
function of n , due to the fact that
n  m mod 3h+1 implies that
3h+1 as a
nk  mk mod 3h+1
k (h, n) depends only on the residue of n mod 3h+1
We shall denote by Nkj(h) the number of k (h, n) which are equal to j
In fact
within a period (j = 0, 1, 2), that is
3h+1-1
Nkj(h) =  k (h, i)
i=0
The values of the ratio Nkj(h)/ 3h+1 (which are obviously equal to 1/3
for k =1 (for any h) and also for h = 0 and for k odd) can be listed as
follows - denoting by Nkj(h) the ratio Nkj(h)/ 3h+1 :
1
For k = 2
with h even
N20(h) = 1/3
N21(h) = (1/3) +
with h odd
N20(h) = (1/3) +
N21(h) = N22(h)
1/3(h/2)+1 ; N22(h) = (1/3) - 1/3(h/2)+1
2/3(s+2)
= (1/3) - 1/3(s+2)
where s = [h/2]
where s = [h/2]
denoting with [x] the integer part of x.
For k even and a divisor of h
Nk0(h) = 1/3
Nk1(h) = (1/3) + 1/3(s+1) ; Nk2(h) = (1/3) - 1/3(s+1) where s = [h/k]
For k even and not a divisor of h
If k is a multiple of 3z (with z = 1, 2, 3, ...), with h <= z mod(nk)
Nk0(h) = 1/3 + 2/3(s+1)
where s = [h/k]
(s+1)
Nk1(h) = Nk2(h) = (1/3) - 1/3
where s = [h/k]
For k odd and a divisor of h
Nk0(h) = Nk1(h) = Nk2(h) = 1/3
For k odd and not a divisor of h
If k is a multiple of 3 z with h <= z mod (nk)
Nk0(h) = (1/3) + 1/3(s+1)
Nk1(h) = (1/3) - 1/3(s+1)
Nk2(h) = 1/3
where s = [h/k]
where s = [h/k]
For k even or odd and not a multiple of 3 z or a multiple of
3 z with h > z mod (nk)
Nk0(h) = 1/3 + 2/3(s+2)
where s = [h/k]
(s+2)
Nk1(h) = Nk2(h) = (1/3) - 1/3
where s = [h/k]
*
*
*
Note - A similar theorem for base 2 was published in:
Gross, W.; Vacca, R. Distribution of the figures 0 and 1 in the various
orders of binary representations of nth powers of integers. Math. Comp. 22
1968 423--427.
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