nL;

y r = 6l.Now we have to find the numbers xo and yo with
similar properties.
To find x7., let's square xo _ , and take the last k digits
of this square/ so thatx?_ r = 10ka +xo. Let's show that
*i - *t is divisible by 10k:
,?-rt =l"?_,-lokalz-("?_, - loka)
= *X_r-2axl_, . 100 + IO*a2 -*?_r+ lOka
: (" ? _ r- xk _, )(r?_, * x1_ 1)
+ 10kl 10ka2 + a - Zax f; - 1lr.
But xo2_ r - xft - r is divisible by lG- 1, and x | _, + xo _, is
divisible by 10 (since both xo_, and xo2_, end in 5).
Therefore, both terms in the last expression are divisible
by 10k.
The construction of yo is a bit more complicated: we
have to take the last k digits of yf _ r. the divisibility
of yk2 - lr7. by 10k is proved in almost the same way:
v? -vx:bf -r-tokblz -(vf -1- lokbl
= b ? -, - vt )lv f -, + Y I - r + Y f - r+ rf- r
+ v f - tl+
lOk c,
where c:tOkb2 +b-zbyf_,. Thetermyos_ r+ ...+yf_r
is divisible by 10 because it's the sum of five numbers
ending in6, andthe differencey?_r-yt _, is divisible
by 1Or - 1
by the induction hypotheiis. So yo2 - yo is divisible
by 10k, and we're done.
Implim[ions
From the first proof, we can see that the sum of the
third and fourth numbers we've found is equal to 10& +1
(indeed, 25 + 76 = 101, 625 + 376: 1,001).
Exercise 1. Prove that this is true for allk.
The second proof demonstrates one arnazing fact:
the fourk-digit endings that are preserved under squaring
are obtained from the respective (k - 1)-digit endings
preserved under squaring merely by adding one
digit on the left! Indeed, the lastk - 1 digits ofxo coincide
with those of x|_r, and so they constitute xo_ r;
only the kth digrt from the right in xo is new. In a similar
argument for the y's we must use the fact that yo _,
comprises the last k - 1 digits of y f _,: this follows from
the identity y
f - t - y * _ r : (V ? _, - y x _ rl(y x _ r + 1 )( yo2_, + 1 )
and the divisibility of y ? _ r - yt _, by 10k - 1.
You may
have noticed this already in our numerical examples:
0-00-000-...,
1-01-001-...,
5-25-625-...,
6-75-375-....
Thus, in the sequence 5,25, ..., xk_r, xk,... , the
terms are built up by adding digits on the left end; this
is also true of the sequence 6,76, ... ,yk_1, yp, .... If we
don't interrupt this process, it will yield two infinite
"numbers"; with ten digits written out they are
x: ...82t289062s,
Y = ...1787t09376.
We cautiously put the word "numbers" in quotation
marks, but, as we'll see in a while, these infinite sequences
can quite legitimately be granted the status of
genuine, though somewhat unusual, numbers.
A new kind olnumhel'
We need a name for our inIinite-to-the-1eft sequences
of digits ...a4azazar. Let's call them supetnumbers.Il
all the digits in such a sequence starting from a certain
place are zeros, we'll say that this supernumber is an
ordinary number-for instance,
...000132 : t32.
Thus, among supemumbers one can find all ordinary
nonnegative integers, but other "numbers" as well.
Supernumbers can be added and multiplied using the
ordinary digit-by-digit rules (fig. 1). They resemble infinite
decimals, as if they were reflected through the
decimal point, except that because of the "carry," the
rules for performing operations are not reflected. This
makes them an essentially new algebraic object. However,
for ordinary supernumbers these operations are
our usual addition and muitiplication. It's interesting
that supernumbers can be subtracted from one another
in any order, also in the usual digit-by-digit way-see
figure 2. (The relation " greaterf srnaLler" cart't be introduced
for supernumbers so as to agree with the algebraic
operations as it does for ordinary numbers.) In
partiatlar, any supernumber can be subtracted from
zero (the supemumber ...000). Therefore, all ordinary
negative numbers can also be found among
supernumbers. For example, figure 3 illustrates the
equality -132 = ...999858. We can see that "ordinary
negative integers" are simply supernumbers with an
infinite row of 9's to the left.
Addition and multiplication of supernumbers, and
subtraction as well, have the usual properties of these
operations: a + b = b + a, ab = ba, a + (b + cl : (a + bl + c,
a(bcl = lablc, a{b + cl = ab + ac, arld so on. In particular,
for arry supemumberxwe can calculate the supemumber
* - ,, and our theorem immediately tells rs that the
a
Figure 1
.... _.6847
+.... ..4219
..._..t066
......6847
x.... ..4219
......t623
......6847
......3694
...... 7388
7493
......2628
Figure 2
4219
6847
.7372
l8
JUtY/AlJEUSI lgS4