The theory of numbers, by Robert D. Carmichael ...

28

THEORY OF NUMBERS

S. When m and n are two relatively prime positive integers the quotient
(rn+n-i)!
m!n!
as an integer.
9*. If mi and n are positive integers, then each of the quotients
(min)! (2m)!(2n2)
na!(mn!)' in!!n(m+nln)!'
is an integer. Generalize to k integers in,, p,....
Io*. If n=a-+-+pq+-rs where a, 3, p, q, r, s are positive integers, then n!
is divisible by
a! 0! r! p! (q!)P(s!)r.
11*. Show that
(rst)
t! (s!)t(r!)st'
is an integer (r, s, t being positive integers). Generalize to the case of n integers
r, S, t,,....
~ 13. REMARKS CONCERNING PRIME NUMBERS
We have seen that the number of primes is infinite. But
the integers which have actually been identified as prime are
finite in number. Moreover, the question as to whether a large
number, as for instance 2257-I, iS prime is in general very
difficult to answer. Among the large primes actually identified
as such are the following:
261 - I 275. 5 +I, 289- I, 2127 - I.
No analytical expression for the representation of prime numbers has yet been discovered. Fermat believed, though he confessed that he was unable to prove, that he had found such an
analytical expression in
22 +I.
Euler showed the error of this opinion by finding that 641 is a
factor of this number for the case when n= 5.
The subject of prime numbers is in general one of exceeding
difficulty. In fact it is an easy matter to propose problems