The reader is aware that there are prime numbers and composite whole
numbers. Now, 1,111,111 cannot be a prime number, because if it were the
only possible answers would be those proposed by Brother Benjamin and
rejected by Father Peter. Also it cannot have more than two factors, or
the answer would be indeterminate. As a matter of fact, 1,111,111 equals
239 x 4649 (both primes), and since each cat killed more mice than there
were cats, the answer must be 239 cats. See also the Introduction, p. 18.
Treated generally, this problem consists in finding the factors, if any,
of numbers of the form (10^_n_ - 1)/9.
Lucas, in his _L'Arithmetique Amusante_, gives a number of curious tables
which he obtained from an arithmetical treatise, called the _Talkhys_, by
Ibn Albanna, an Arabian mathematician and astronomer of the first half of
the thirteenth century. In the Paris National Library are several
manuscripts dealing with the _Talkhys_, and a commentary by Alkalacadi,
who died in 1486. Among the tables given by Lucas is one giving all the
factors of numbers of the above form up to _n_ = 18. It seems almost
inconceivable that Arabians of that date could find the factors where _n_
= 17, as given in my Introduction. But I read Lucas as stating that they
are given in _Talkhys_, though an eminent mathematician reads him
differently, and suggests to me that they were discovered by Lucas
himself. This can, of course, be settled by an examination of _Talkhys_,
but this has not been possible during the war.
The difficulty lies wholly with those cases where _n_ is a prime number.
If _n_ = 2, we get the prime 11. The factors when _n_ = 3, 5, 11, and 13
are respectively (3 . 37), (41 . 271), (21,649 . 513,239), and (53 . 79 .
265371653). I have given in these pages the factors where _n_ = 7 and 17.
The factors when _n_= 19, 23, and 37 are unknown, if there are any.[B]
When _n_ = 29, the factors are (3,191 . 16,763 . 43,037. 62,003 .
77,843,839,397); when _n_ = 31, one factor is 2,791; and when _n_ = 41,
two factors are (83 . 1,231).
[B] Mr. Oscar Hoppe, of New York, informs me that, after reading my
statement in the Introduction, he was led to investigate the case of _n_
= 19, and after long and tedious work he succeeded in proving the number
to be a prime. He submitted his proof to the London Mathematical Society,
and a specially appointed committee of that body accepted the proof as
final and conclusive. He refers me to the _Proceedings_ of the Society
for 14th February 1918.
As for the even values of _n_, the following curious series of factors
will doubtless interest the reader. The numbers in brackets are primes.
_n_ = 2 = (11)
_n_ = 6 = (11) x 111 x 91
_n_ = 10 = (11) x 11,111 x (9,091)
_n_ = 14 = (11) x 1,111,111 x (909,091)
_n_ = 18 = (11) x 111,111,111 x 90,909,091