Contents
X
Appendices
A.l The Fundamental Theorem of Algebra
482
A.2 Symmetric Functions
484
A.3 A Special Value of the Riemann Zeta Function
A.4 Linear Recurrences
493
482
490
General References
500
Hints
503
Answers
512
Index
522
18
Divisibility
18. Find the values of (a, b) and [a, b] if a and b are positive integers
such that a lb.
19. Prove that any set of integers that are relatively prime in pairs are
relatively prime.
20. Given integers a and b, a number n is said to be of t
34
Divisibility
*48. Prove that there are infinitely many primes by considering the se24
22
23
quence 2 21 + 1,2 + 1,2 + 1,2 + 1, .(H)
*49. If g is a divisor of each of ab, cd, and ac + bd, prove that it is also a
divisor of ac and bd, where a, b, c, d ar
1. 2
Divisibility
7
Definition 1.2 The integer a is a common divisor of band c in case alb and
a Ic. Since there is only a finite number of divisors of any nonzero integer,
there is only a finite number of common divisors of b and c, except in the case
b
1.3
23
Primes
place, if we had a factoring 5 = (x 1 + y 1 v'=-6Xx 2 + y 2 v'=-6) into complex numbers, we could take norms to get
and a similar argument
which contradicts (1.2). Thus, 5 is a prime in
establishes that 2 is a prime.
We are now in a position
1.3
21
Primes
n 1n 2 , where 1 < n 1 < n and 1 < n 2 < n. If n 1 is a prime, let it stand;
otherwise it will factor into, say, n 3 n 4 where 1 < n 3 < n 1 and 1 < n 4 < n 1;
similarly for n 2 . This process of writing each composite number that arises
as
5
1.2 Divisibility
Theorem 1.1
(J) alb implies a Ibe for any integer c;
(2) alb and blc imply a lc;
(3) alb and a lc imply a l(bx
+ cy) for any integers x andy;
(4) alb and bla imply a= b;
(5) alb, a
> 0, b > 0, imply a
b;
(6) if m =F 0, alb implies and i
62
Congruences
Similarly, we say that f(x) = 0 (mod m) is an identical congruence if it
holds for all integers x. If f(x) is a polynomial all of whose coefficients are
divisible by m, then f(x) = 0 (mod m) is an identical congruence. A
different type of i
Notation
Items are listed in order of appearance.
7L
Q
IR
alb
a,f'b
akllb
[x]
(b,c)
gcd(b, c)
(bl, bz, ' bn)
[at, az, an]
N(a)
1T(X)
f(x)- g(x)
MP
n
E
.A/
(%)
tif(x)
a= b (mod m)
a' b (mod m)
The set of integers, 4
The set of rational numbers, 4
The set o
CHAPTER2
Congruences
2.1
CONGRUENCES
It is apparent from Chapter 1 that divisibility is a fundamental concept of
number theory, one that sets it apart from many other branches of
mathematics. In this chapter we continue the study of divisibility, but from
2.1
53
Congruences
pl(x - 1Xx + 1). By Theorem 1.15 it follows that pl(x - 1) or pl(x + 1).
Equivalently, x = 1 (mod p) or x = -1 (mod p). Conversely, if either one
of these latter congruences holds, then x 2 = 1 (mod p).
Theorem 2.11 Wilson's theorem. If
1.2
17
Divisibility
PROBLEMS
1. By using the Euclidean algorithm, find the greatest common divisor
(g.c.d.) of
(b) 2689 and 4001;
(a) 7469 and 2464;
(c) 2947 and 3997;
(d) 1109 and 4999.
2. Find the greatest common divisor g of the numbers 1819 and 3587,
Divisibility
8
Proof Part 1 follows from the proof of Theorem 1.3. To prove part 2, we
observe that if d is any common divisor of b and c, then d lg by part 3 of
Theorem 1.1. Moreover, there cannot be two distinct integers with property 2, because of Theo
30
Divisibility
19. Let a and b be positive integers such that (a, b) = 1 and ab is a
perfect square. Prove that a and b are perfect squares. Prove that
the result generalizes to kth powers.
20. Given (a, b, c)[a, b, c] =abc, prove that (a, b)= (b, c) = (
2.1
51
Congruences
Theorem 2.5 The number </J(m) is the number of positive integers less than
or equal to m that are relatively prime to m.
Euler's function </J(m) is of considerable interest. We shall consider it
further in Sections 2.3, 4.2, 8.2, and 8.
Contents
viii
3.5
Equivalence and Reduction of Binary Quadratic
Forms
155
163
3.6 Sums of Two Squares
3.7 Positive Definite Binary Quadratic Forms
170
Notes on Chapter 3
176
4
Some Functions of Number Theory
4.1
4.2
4.3
4.4
4.5
5
212
The Equation ax + by
1. 2
19
Divisibility
36. Prove that (a, b, c) = (a, b), c).
37. Prove that (a 1, a 2 ,- ,an)= (a 1, a 2 ,- , an_ 1), an).
38. Extend Theorems 1.6, 1.7, and 1.8 to sets of more than two integers.
39. Suppose that the method used in the proof of Theorem 1.1
An Introduction to
the Theory of Numbers
FIFTH EDITION
Ivan Niven
University of Oregon
Herbert S. Zuckerman
University of Washington
Hugh L. Montgomery
University of Michigan
John Wiley & Sons, Inc.
New York Chichester Brisbane Toronto Singapore
1.4
41
The Binomial Theorem
and 'Y be disjoint sets containing m and n elements,
u 'Y. Show that the number of
respectively, and put ./=
subsets N of ._/ that contain k elements and that also have the
contains elements is cfw_
k
Interproperty that N n
pre
14
Divisibility
Remark on Calculation. We note that X; is determined from X;_ 1 and
x;_ 2 by the same formula that r; is determined from r;_ 1 and r;_ 2 That is,
X;
=
Y;
= Y;-2
X;_ 2 -
Q;X;-p
and similarly
- Q;Y;-I
The only distinction between the three s
7.6
7.7
7.8
7.9
8
Primes and Multiplicative Number Theory
8.1
8.2
8.3
8.4
9
Best Possible Approximations
341
Periodic Continued Fractions
344
Pell's Equation
351
Numerical Computation
358
Notes on Chapter 7
359
360
Elementary Prime Number Estimates
360
Di
2.1
59
Congruences
*48. If r 1,r2 ,.,rP and
*49.
*50.
51.
*52.
53.
54.
*55.
are any two complete residue systems modulo a prime p > 2, prove that the set r 1rf,r2 r2,.
cannot be a complete residue system modulo p.
If p is any prime other than 2 or 5, pro
2.2
61
Solutions of Congruences
are automatically solutions, so this entire congruence class is counted as a
single solution.
Definition 2.4 Let r 1, r 2 , , rm denote a complete residue system modulo
m. The number of solutions of f(x) = 0 (mod m) is the
38
Divisibility
form described in Definition 1.6. Thus the two proofs of Theorem 1.22
may be combined to provide a second proof of Theorem 1.20.
As a matter of logic, we require only one proof of each theorem, but
additional proofs often provide new insig
50
CongnJences
division by m; thus a = qm + r by Theorem 1.2. Now a = r (mod m) and,
since r satisfies the inequalities 0 .,; r < m, we see that every integer is
congruent modulo m to one of the values 0, 1, 2, , m - 1. Also it is clear
that no two of the
vi
Preface
been introduced. We address a number of calculational issues, most
notably in Section 1.2 (Euclidean algorithm), Section 2.3 (the Chinese
remainder theorem), Section 2.4 (pseudoprime tests and Pollard rho
factorization), Section 2.9 (Shanks' RE
60
57.
58.
59.
60.
Congruences
pairs (u, v ), show that at least one of the equations a 2 + 2b 2 = p,
a 2 + 2b 2 = 2p has a solution.
Show that (a + bF'iXc + dF'f) = (ac - 2bd) + (be +
Thus or otherwise show that (a 2 + 2b 2 Xc 2 + 2d 2 ) = (ac
- 2bd) 2 +
Preface
This text is intended for use in a first course in number theory, at the
upper undergraduate or beginning graduate level. To make the book
appropriate for a wide audience, we have included large collections of
problems of varying difficulty. Some
1. 2
9
Divisibility
Proof The second assertion is the special case of the first obtained by
using the greatest common divisor g of a and b in the role of d. The first
assertion in turn is a direct consequence of Theorem 1.6 obtained by
replacing m, a, b i
24
Divisibility
where the factors P; and qj are primes, not necessarily all distinct, but
where no prime on the left side occurs on the right side. But this is
impossible because PIIqiq 2 q 5 , so by Theorem 1.15, pi is a divisor of
at least one of the %
20
Divisibility
plane. Similarly, draw n - 2 additional regular pentagons on the
base sides P 1P 3 , P 1P4 ,. , P 1Pn, all pentagons lying on the same
side of the line P 1Pn. Mark dots at each vertex and at unit intervals
along the sides of these pentagon