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

36

THEORY OF NUMBERS

EXERCISES
I. Show that the indicator of any integer greater than 2 is even.
2. Prove that the number of irreducible fractions not greater than I and with
denominator equal to n is ~(n).
3. Prove that the number of irreducible fractions not greater than i and
with denominators not greater than n is
~(I)+0(2)+~(3)+... +(n).
4. Show that the sum of the integers less than n and prime to n is ~nO(n)
if n>I.
5. Find ten values of x such that ~(x) = 24.
6. Find seventeen values of x such that 0(x)= 72.
7. Find three values of 1 for which there is no x satisfying the equation
~(x) = 2n.
S. Show that if the equation
~(x) =n
has one solution it always has a second solution, n being given and x being the
unknown.
9. Prove that all the solutions of the equation
~(x) =4n-2, n'>I,
are of the form pa and 2p", where p is a prime of the form 4s-I.
io. How many integers prime to n are there in the set
a) 1 2, 2.3, 3'4,.. n(n+i)?
l) 1-2.3, 2-3-4, 3-'45,..., n(n4-I)(n+2)?
1 2 2'3 3'4 n(n+I)....v
2 2 2 2
1-2.3 2-3'4 3'4'5 n(n+I)(n+2)
d) ~, ~ -, -,..., -
6 6 6 6
rI*. Find a method for determining all the solutions of the equation
+(x) =n,
where n is given and x is to be sought.
12*. A number theory function f(n) is defined for every positive integer n;
and for every such number n it satisfies the relation
~(d) + (d)+... + (dr)n,
where di, d2,.., dr are the divisors of n. From this property alone show
that
w(n)=her p, pe f s o
where pi, p2,..., p are the different prime factors of nl.