A.l The Fundamental Theorem of Algebra
A.2 Symmetric Functions
A.3 A Special Value of the Riemann Zeta Function
A.4 Linear Recurrences
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
20. Given integers a and b, a number n is said to be of t
*48. Prove that there are infinitely many primes by considering the se24
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
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
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
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
(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
(6) if m =F 0, alb implies and i
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
Items are listed in order of appearance.
(bl, bz, ' bn)
[at, az, an]
a= b (mod m)
a' b (mod m)
The set of integers, 4
The set of rational numbers, 4
The set o
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
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. By using the Euclidean algorithm, find the greatest common divisor
(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,
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
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) = (
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.
Equivalence and Reduction of Binary Quadratic
3.6 Sums of Two Squares
3.7 Positive Definite Binary Quadratic Forms
Notes on Chapter 3
Some Functions of Number Theory
The Equation ax + by
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
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
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_
Interproperty that N n
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;_ 2 -
The only distinction between the three s
Primes and Multiplicative Number Theory
Best Possible Approximations
Periodic Continued Fractions
Notes on Chapter 7
Elementary Prime Number Estimates
*48. If r 1,r2 ,.,rP and
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
Solutions of Congruences
are automatically solutions, so this entire congruence class is counted as a
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
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
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
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