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50 THE FASCINATION OF NUMBERS

A special interest has been taken over the years in testing
for primality those numbers which consist entirely of a
repetition of the digit 1. For a long time, the number 11 was
the only such number known for certain to be prime, but
in 1918 a proof was claimed that the number represented by
nineteen digits was also a prime. No other prime of this form
has been discovered.

In cases where n (=number of digits) is even, the number
is obviously not prime since it will have the factor 11. For
cases where 7 is odd, some factors are known. Where 7 equals
3, 5 and 7, the factors are respectively:

Il 13 X837
AL,XTT AT WAy
I,II1,I11I=239 X4649
On the other hand, the factors, if any, where n equals 23 and
37 are unknown.

J 10" —1
Numbers of this nature are of the form ,» and are, as

 

such, recognizable as being associated with the cycles of
recurring decimals (see Chapter 10), and this knowledge
enables us to reduce the amount of work in finding their
factors. It will be found that the number of digits in a recur-
ring decimal cycle is related to the prime number which,
when used as the divisor of unity, generates the cycle. In
fact, the number of digits in a cycle is either (x—1) or a
factor of (x —1) where « is the prime divisor other than 2 or 5.
Now, a number consisting entirely of nines is obviously nine
times greater than a number consisting of an equal number
of ones, so that any factors of 999 (other than g itself) must
also be factors of 111. Therefore, in order to find the factors
of 111, we can instead find the factors of 999, and the work of
finding the latter is simplified by the knowledge that a num-
ber of n nines will have a divisor (if any) equivalent to a
multiple of # added to 1.

Thus, the factors of 11,111, where n=F5, are each greater
by 1 than multiples of 5. They are 41 and 271. The multiples