2-adic Summation

The 2-adic representation of an integer

Just as each non-negative integer m has a "base 10" representation

m = d0 + 10 d1 + 102 d2 + ... ,

where d0 is the unit's digit, d1 the ten's digit, etc., with dj a non-negative integer no larger than 9 = 10 - 1, it is likewise the case for any given base b > 1 that every non-negative integer m has a base b representation, also called its b-adic representation,

m = d0 + b d1 + b2 d2 + ... ,

where each dj is a non-negative integer no larger than b-1.

In particular, every non-negative integer m has a 2-adic representation

m = d0 + 2 d1 + 22 d2 + ... ,

where each dj is either 0 or 1.

The existence of the 2-adic representation of any postive integer is equivalent to the statement that every positive integer is the sum of distinct powers of 2 (provided that 1 = 20 is included as a power of 2).

We shall focus on the 2-adic representation of m.

The 2-adic distance between two integers

Let us define the 2-adic order ord2(m) of m by the formula

ord2(m) = the least exponent of 2 in the 2-adic representation of m .

This definition only makes sense when m is a positive integer. When m is a negative integer, we define ord2(m) = ord2(-m).

In other words e = ord2(m) if m is divisible by 2e but not by any higher power of 2. Viewed this way it makes sense to say that ord2(0) is infinite since 0 is divisible by each power of 2.

We define the 2-adic norm ||m|| of any integer m by the formula

||m|| = 2-e   when   e = ord2(m) .

with the understanding that ||0|| = 0.

Then we define the 2-adic distance d(m, n) between two integers m and n by the predictable formula

d(m, n) = || m - n || .

The geometric series summed 2-adically

Consider the infinite series

1 + 2 + 22 + 23 + 24 + ... .

The partial sum sn after n terms is just the 2-adic expansion of the integer 2n - 1. That is,

sn = 1 + 2 + 22 + 23 + ... + 2n-1 = 2n - 1 .

Since ||2n|| = 2-n approaches 0 as n becomes large, it follows that this geometric series -- the sum of the powers of 2 -- converges to the sum -1 with respect to 2-adic distance.

To say the least, it is curious that the 2-adic method of summing all of the powers of 2 gives the same result as the method of analytic continuation.

One must bear in mind that the two contexts are completely different. The notion of 2-adic summation rests on the notion of 2-adic limit, which, in turn, rests on the notion of 2-adic distance.

The 2-adic distance d(m, n) has been defined above only when m and n are integers. Every non-negative integer has a finite 2-adic expansion, and we have just seen that the integer -1 has an infinite 2-adic expansion as the sum of all powers of 2 when infinite sums are interpreted 2-adically.

2-adic integers

If we introduce the term 2-adic integer to mean any infinite expansion (at first without regard to convergence)

f = c0 + 2 c1 + 22 c2 + ... ,

where each cj is either 0 or 1, then we can define

ord2(f) = the least j with cj = 1 ,

and then

||f|| = 2-e   when   e = ord2(f)

and

d(f, g) = || f - g || .

It is then obvious that any 2-adic integer is the limit, for 2-adic distance, of its sequence of partial sums. Consequently, every 2-adic integer is the limit of a sequence of positive integers.

Moreover, it can be shown that every negative integer is a 2-adic integer and that every ratio a/b of integers a and b with b odd is a 2-adic integer.

Needless to say, the set of 2-adic integers is not an identifiable subset of the set of real numbers.

But there is a nearly one-to-one correspondence of the set of 2-adic integers with the closed interval [0, 1] of all real numbers between 0 and 1. This correspondence is obtained by associating to the 2-adic integer

f = c0 + 2 c1 + 22 c2 + ...

the real number

R(f) = 2-1 c0 + 2-2 c1 + 2-3 c2 + ... .

One reason for interest in the set of 2-adic integers is that it has a "fractal" nature.

Further insight into 2-adic analysis may be obtained by pursuing the analogous study of formal power series.


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