Elementary Number Theory Chapter 1 Divisibility Section 1.0 Introduction
Euclid (350 B.C.): There are infinitely many prime numbers.
Euclid: √
2 is an irrational number.
J. L. Lagrange(1736-1813): (Four Square Theorem) Every natural number is expressible as a sum of four squares, and it is best possible.
Pierre de Fermat (1601-1665): (Fermat Last Theorem) The equation xn+ yn = zn has no solutions in positive integers for any exponent n > 2.
Leonhard Euler(1707-1783): (Conjecture) No nth power is a sum of fewer than n nth powers.
True for n = 3.
Counterexample: L.J.Lander and Thomas Parkin, (1968) 1445 = 275 + 845 + 1105 + 1335; N.J.Elkies, (1987) 206156734 = 26824404 + 153656394 + 187967604.
Christian Goldbach Conjecture (1742): Every even integer greater than 2 is a sum of two primes.
John Wilson (1741-1793): every prime p is a divisor of (p− 1)! + 1. First proof was given by Lagrange.
3n + 1 conjecture: Define T (n) =
{ 3n+1
2 for n odd
n
2 for n even . Then, for any integer n > 1, the sequence T (n), T (T (n)), T (T (T (n))), . . . eventually reaches 1.
Triangular Numbers. Each of the numbers 1 = 1, 3 = 1 + 2, 6 = 1 + 2 + 3,· · · represents the number of dots that can be arranged evenly in an equilateral triangle. The ancient Greeks called these triangular numbers.
(a) A number is triangular if and only if it is of the form n(n+1)2 . (Pythago- ras, 550 B.C)
(b) A number n is triangular if and only if 8n + 1 is a perfect square.
(Plutarch 100 A.D.)
(c) The sum of any two consecutive triangular number is a perfect square.
(Nicomachus 100 A.D.)
(d) If n is a triangular number, then so is 9n + 1, 25n + 3, 49n + 6. (Euler, 1775)
1
2
Pentagonal numbers. Each of the numbers 1 = 1, 5 = 1 + 4, 12 = 1 + 4 + 7,· · · represents the number of dots that can be arranged evenly in an pentagon. The ancient Greeks called these pentagonal numbers. If pn is the nth pentagonal number, then pn = n(3n2−1).
Section 1.1 Divisibility
Axiom 1.1.1 (Well-Ordering Principle) Every nonempty set of nonnegative integers contains a least element.
Theorem 1.1.2 (Archimedean property) If a and b ∈ N, then there exists a positive integer n such that na > b.
Theorem 1.1.3 (First principle of induction) Let S be a set of positive integers with the following properties:
(i) 1∈ S;
(ii) whenever k∈ S, the integer k + 1 must also be in S.
Then S = N.
Theorem 1.1.4 (Second principle of induction) Let S be a set of positive inte- gers with the following properties:
(ii) 1 ∈ S;
(ii) If k is an positive integer such that 1, 2, . . . , k belong to S, then the integer k + 1 must also be in S.
Then S =N.
Example 1.1.5 (a) 12 + 22+· · · + n2 = n(2n+1)(n+1)
6 .
(b) 1 + 2 + 22 +· · · + 2n−1 = 2n
(c) The Lucas sequence is defined by a1 = 1, a2 = 3, an = an−1+an−2, for all n ≥ 3. Then an < (74)n for all n.
Definition 1.1.6 (a) Let a, b ∈ Z, a 6= 0. b is divisible by a if there is x ∈ Z such that b = ax, and we write a|b. In case b is not divisible by a, we write a 6 | b.
(b) If a|b and 0 < a < b, then a is called a proper divisor of b.
(c) akkb is used to indicate that ak|b but ak+1 6 | b.
Lemma 1.1.7 (a) a|b ⇒ a|bc for any c ∈ Z.
(b) a|b and b|c ⇒ a|c.
(c) a|b and b|c ⇒ a|(bx + cy) for any x, y ∈ Z.
(d) a|b and b|a ⇒ a = ±b.
(e) a, b ∈ N, a|b ⇒ a ≤ b.
(f) If m6= 0, a|b ⇔ ma|mb.
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Proposition 1.1.8 (The division algorithm) Given a, b ∈ Z, b > 0, there exist unique integers q, r such that a = qb + r, 0 ≤ r < b. If b 6 | a then r satisfies 0 < r < b.
Proposition 1.1.9 (Other version of the division algorithm) Given a, b∈ Z, b >
0, there exist unique integers q, r such that a = qb + r, where −12b < r ≤ 12b. If b6 | a then r satisfies 0 < |r|.
Example 1.1.10 a(a23+2) is an integer for all a∈ Z.
Definition 1.1.11 (a) The integer a is a common divisor of b, c in case a|b and a|c.
(b) (There are only finite number of common divisors of b and c, except b = c = 0.) If at least one of b and c is not 0, the greatest among their common divisors is called the greatest common divisor of b and c and is denoted by (b, c).
We denote the greatest common divisor of b1, b2, . . . , bn, not all zero, by (b1, b2, . . . , bn).
Proposition 1.1.12 If g = (b, c), then there exist integers x, y such that g = bx + ay.
Proposition 1.1.13 The following statements are equivalent:
(a) g = (b, c).
(b) g is the least positive value of bx + cy where x and y range over all integers.
(c) g is the positive common divisor of b and c that is divisible by every common divisor.
Remark 1.1.14 An integer d is expressible in the form d = bx + cy, then d is not necessarily the g.c.d of b and c. But (b, c) is a divisor of d. So, if bx + cy = 1, then (b, c) = 1.
Proposition 1.1.15 Given b1, b2, . . . , bn, not all zero, there exist integers x1, x2, . . . , xn such that g = (b1, b2, . . . , bn) = ∑n
j=1bjxj. g is the least positive value of the linear form ∑n
j=1bjyj where the yj range over all integers; also g is the positive common divisor of b1, b2, . . . , bn and c that is divisible by every common divisor.
Lemma 1.1.16 (a) (ma, mb) = m(a, b) for any m ∈ Z.
(b) If d > 0 is a common divisor of a and b, then (ad, bd) = 1d(a, b).
(c) If (a, m) = (b, m) = 1, then (ab, m) = 1.
Definition 1.1.17 (a) We say that a1, a2, . . . , an are relatively prime in case (a1, a2, . . . , an) = 1.
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(b) We say that a1, a2, . . . , an are relatively prime in pairs in case (ai, aj) = 1 for all i6= j.
Proposition 1.1.18 (a, b) = (a,−b) = (a, b + ax) for all x ∈ Z.
Theorem 1.1.19 If c|ab and (b, c) = 1, then c|a.
Theorem 1.1.20 (The Euclidean Algorithm) Given b ∈ Z, c ∈ N, we obtain a series of equations
b = cq1 + r1, 0 < r1 < c, c = r1q2 + r2, 0 < r2 < r1, r1 = r2q2+ r3, 0 < r3 < r2,
· · · ·
rj−2 = rj−1qj + rj, 0 < rj < rj−1, rj−1 = rjqj+1.
(b, c) = rj and values of x, y in (b, c) = bx + cy can be obtained by writing each ri as a linear combination of b and c.
Remark 1.1.21 (a) Gabriel Lam´e (French, 1795-1870) prove that the number of steps required in the Euclidean Algorithm is at most five times the number of digit in the smaller integer, that is, the number of iterations j of the Euclidean algorithm has an upper bound j < 3 ln c.
(b) For most pairs b, c, it requires approximately 12 ln 2π2 ln c steps. [J.
Dixon(1970)]
(c) For any n > 0, it is possible to find an, bn such that exactly n divisions are required to compute gcd.
(d) The number of steps usually cab be reduced by selecting remainders rk+1 such that |rk+1| < |r2k|. [Example:12378, 3054]
Example 1.1.22 (a) Find the greatest common divisor of 42823 and 6409.
(b) Find integers x and y to satisfy 42823x + 6409y = 17. (Two Methods) (c) Find g = (b, c) where b = 5033464705 and c = 3137640337, and deter- mine x, y such that bx + cy = g.
Definition 1.1.23 The integers a1, a2, . . . , an, all different from zero, have a common multiple b if ai|b for all i. The least of the positive common multiples is called the least common multiple, and is denoted by [a1, a2, . . . , an].
Theorem 1.1.24 If b is any common multiple of a1, a2, . . . , an, then [a1, a2, . . . , an]|b.
Theorem 1.1.25 If m > 0, [ma, mb] = m[a, b]. Also [a, b](a, b) = |ab|.
