Elementary Number Theory Section 1.2 Primes
Definition 1.2.1 An integer p > 1 is called a prime in case there is no divisor d of p satisfying 1 < d < p. If an integer a < 1 is not a prime, it is called a composite number.
Theorem 1.2.2 Every integer n > 1 can be expressed as a product of primes.
Lemma 1.2.3 If p|a1a2· · · an, p being a prime, then p divides at least one factor ai.
Theorem 1.2.4 (Fundamental theorem of arithmetic) The factorization of any integer n > 1 into primes is unique apart from the order of the prime factors.
[Two proofs.]
Remark 1.2.5 (a) For any positive integer a, a =∏
ppα(p)is called the canonical factoring of n into prime powers.
(b) let a = ∏
ppα(p), b = ∏
ppβ(p), c = ∏
ppγ(p). If c = ab, then γ(p) = α(p) + β(p).
(a, b) = ∏
ppmin(α(p),β(p)), [a, b] =∏
ppmax(α(p),β(p)). a is a perfect square if and only if, for all p, α(p) is even.
(c) The second proof of 1.2.4 is independent of the previous theorems, so the formulas of (a, b), [a, b] can be used to prove many results in Section 1.1.
Example 1.2.6 The number systems in which the factorization is not unique.
(a) E = {2, 4, 6, 8, · · · }.
(b) C = {a + b√
6 : a, b∈ Z}.
Example 1.2.7 (Pythagoras) The number √
2 is irrational.
1.2.8 The Sieve of Eratosthnes (276-194 B.C.) Write down the integers from 2 to n in natural order and then systematically eliminate all the composite numbers by striking out all multiples of p of the primes p ≤ √
n. The integers that left on the list are primes.
Theorem 1.2.9 (Euclid) The number of primes is infinite. [Three proofs.]
Remark 1.2.10 It is not known whether there are infinitely many prime p for which p#+ 1 is also prime, where p# is the product of all primes that less than or equal to p.
At present, 19 primes of the form p# + 1 have been identified: p = 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801,
1
2
24029, 42209 (discovered in 2000). The integerp#+ 1 is composite for all other p ≤ 120000.
Remark 1.2.11 Let pn denote the nth of the prime numbers in natural order.
(a) pn < p1p2· · · pn−1+ 1, n≥ 2.
(b) pn < p1p2· · · pn−1− 1, n ≥ 3.
(c) (Bonse inequality) p2n < p1p2· · · pn−1, n≥ 5.
(d) p2n ≤ p2p3· · · pn− 2, n ≥ 3
Theorem 1.2.12 If pn is the nth prime number, then pn ≤ 22n−1.
Corollary 1.2.13 For n ≥ 1, there are at least n + 1 primes less than 22n. Theorem 1.2.14 (Bertrand conjecture 1845, proved by P.L. Tchebycheff in 1852) Between n ≥ 2 and 2n there is at least one prime.
Corollary 1.2.15 pn < 2n.
Theorem 1.2.16 For every real number y ≥ 2 , ∑
prime p≤y 1
p > ln ln y− 1.
Remark 1.2.17 (a) A corollary of 1.2.16 is 1.2.9.
(b) It can be shown that ∑
prime p≤y 1
p− ln ln y is a bounded function of y.
(c) (Prime Number Theorem) limx→∞ x/ ln xπ(x) = 1, that is π(x)∼ ln xx , where π(x) is the number of primes ≤ x.
Remark 1.2.18 A repunit is an integer written as a string of 1’s. Let Rn denote the repunit consisting of n consecutive 1’s. R2, R19, R23, R317, R1031, R49081, R86453( discovered in 2001) are primes. These are the only possible Rn for all n≤ 45000.
Conjecture 1.2.19 (a) There are infinitely many primes of the form n2− 2.
(b) There are infinitely many primes of the form 2n+ 1.
(c) There are infinitely many primes of the form n2+ 1.
(d) There are infinitely many primes of the form 2n− 1.
(e) There are infinitely many primes p such that p + 50 is also prime.
(f) Every even integer can be written as the difference of two consecutive primes in infinitely many ways.
Remark 1.2.20 There is an unsolved question: Whether there are infinitely many pairs of twin primes.
The largest twins are 33218925· 2169690± 1 (discovered in 2002).
Theorem 1.2.21 There are arbitrarily large gaps in the series of primes.
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Remark 1.2.22 (a) The largest gap discovered is 1132 after the prime 1693182318746371.
(b) Conjecture: There is a prime gap for every even integer.
Remark 1.2.23 (a) Goldbach Conjecture (1972): Every even integer is the sum of two numbers that are either primes or 1.
(b) More generally, every even integer greater than 4 is the sum of two odd prime numbers.
(c) This conjecture implies that each odd number larger than 7 is a sum of three odd primes.
(d) It is known that every even integer is a sum of six or fewer primes.
Theorem 1.2.24 (Hardy, Littlewood, 1922; I.M. Vinogradov, 1937; Borozdkin, 1956; 2002) All odd integers large than 101346 can be written as a sum of three odd primes.
Proposition 1.2.25 There are infinitely many primes of the form 4n + 3.
Theorem 1.2.26 (P.G.L. Dirichlet,1837) If a and d are relatively prime positive integers, then the arithmetic progression a, a + d, a + 2d, a + 3d, . . . contains infinitely many primes.
Theorem 1.2.27 If all the terms of the arithmetic progression p, p+d, p+2d, p+
3d, . . . , p + (n− 1)d are prime numbers, then d is divisible by every prime q < n.
Remark 1.2.28 (a) There is an unsolved problem: Whether there exist arbi- trary long arithmetic progression consists only of primes.
(b) The longest progression found to date is 114103378550553+4609098694200n, 0≤ n ≤ 21.
(c) A sequence of 10 consecutive primes which is an arithmetic progression was discovered, the common difference is 210.
Remark 1.2.29 (a) Let f (n) = n2 + n + 41. Then f (k) are primes for k = 0, 1, 2, . . . , 40.
(b) Let g(n) = 103n2 − 3945n + 34381. Then g(k) are primes for k = 0, 1, 2, . . . , 42.
(c) Let h(n) = 36n2 − 810n + 2753. Then h(k) gives a string of 45 prime values.
Lemma 1.2.30 It is impossible to find a polynomial f (n) such that f (k) are primes for all k∈ N.
Theorem 1.2.31 (W.H.Mills,1947) There is a positive real number r such that f (n) =br3nc is prime for n = 1, 2, 3, . . . .
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Section 1.3 The Binomial Theorem
Definition 1.3.1 Let α ∈ R, and k ∈ N. Then the binomial coefficient (α
k
) is given by α(α−1)···(α−k+1)
k! .
Lemma 1.3.2 The product of any k consecutive integers is divisible by k!.
Theorem 1.3.3 (The binomial Theorem) For any integer n ≥ 1 and any real numbers x, y, (x + y)n =∑n
k=0
(n
k
)xkyn−k. [Combinatoric proof, analytic proof]
Lemma 1.3.4 Let P (z) =∑n
k=0akzk ∈ C[z]. Then ar = P(r)r!(0) for 0≤ r ≤ n.
Lemma 1.3.5 (n
k
)+(n+1
k
)=(n+1
k+1
)for n, k ∈ N.
Theorem 1.3.6 (1 + z)α = ∑∞
k=0
(α
k
)zk for |z| < 1. [Combinatoric proof, analytic proof]
Example 1.3.7 The Catalan numbers defined by Cn = n+11 (2n
n
), n≥ 0. It first appeared in 1938 when Eugene´ene Catalan(1814-1894) show that there are Cn ways of parenthesizing a nonassociative product of n + 1 factors.
