the little book of bigger primes - ribenboim

[...]... Recognize Whether a Natural Number is a Prime 15 I The Sieve of Eratosthenes 16 xii Contents II Some Fundamental Theorems on Congruences A Fermat’s Little Theorem and Primitive Roots Modulo a Prime B The Theorem of Wilson C The Properties of Giuga and of Wolstenholme D The Power of a Prime Dividing a Factorial E The Chinese Remainder Theorem ... product of primes But the series ∞ (1/n) is divergent; being a series of positive n=1 terms, the order of summation is irrelevant, so the left-hand side is infinite, while the right-hand side is clearly finite This is absurd In Chapter 4, I will return to developments along this line IV Thue’s Proof Thue’s proof uses only the fundamental theorem of unique factorization of natural numbers as products of prime... gcd(a, d) = 1} Linnik’s constant set of all k-almost -primes Schnirelmann’s constants number of representations of 2n as sum of two primes #{2n | 2n ≤ x, 2n is not a sum of two primes nth pseudoprime number of pseudoprimes to base 2, less than or equal to x same, to base a number of Euler pseudoprimes to base 2, less than or equal to x same, to base a number of strong pseudoprimes to base 2, less than or... prime, then ∞ k=0 1 1 = 1 − (1/q) qk Multiplying these equalities: 1+ 1 1 1 1 1 1 1 + + + × + + ··· = p q p2 pq q 2 1 − (1/p) 1 − (1/q) Explicitly, the left-hand side is the sum of the inverses of all natural numbers of the form ph q k (h ≥ 0, k ≥ 0), each counted only once, because every natural number has a unique factorization as a product of primes This simple idea is the basis of the proof IV... multiple of the integers m, n natural logarithm of the real number x > 0 ring of integers field of rational numbers field of real numbers field of complex numbers The following notations are listed as they appear in the book: xviii Index of Notations Page Notation Explanation 3 4 pn p# 6 15 Fn [x] 19 28 29 30 30 gp ϕ(n) λ(n) ω(n) L(x) 31 34 35 36 36 Vϕ (m) t∗ n k(m) P [m] Sr the nth prime product of all primes. .. the fundamental theorem: There exist infinitely many prime numbers I shall give several proofs of this theorem (plus four variants), by famous, but also by forgotten, mathematicians Some proofs suggest interesting developments; others are just clever or curious There are of course more (but not quite infinitely many) proofs of the existence of infinitely many primes I Euclid’s Proof Suppose that p1 = 2... 174 E The Nontrivial Zeros of ζ(s) 177 F Zero-Free Regions for ζ(s) and the Error Term in the Prime Number Theorem 180 G Some Properties of π(x) 181 H The Distribution of Values of Euler’s Function 183 II The nth Prime and Gaps Between Primes 184 A The nth Prime 185 B Gaps Between Primes 186 III Twin Primes ... Facing the task of presenting the records on prime numbers, I was led to think how to organize this volume In other words, to classify the main lines of investigation and development of the theory of prime numbers It is quite natural, when studying a set of numbers—in this case the set of prime numbers—to ask the following questions, which I phrase informally as follows: How many? How to decide whether... ln+1 is the smallest prime factor of l1 l2 · · · ln + 1 Shanks conjectured that every prime belongs to the sequence, but the truth of this assertion is still undecided Wagstaff (1993) computed all terms ln for n ≤ 43, continuing previous calculations by Guy & Nowakowski (1975) 6 1 How Many Prime Numbers Are There? The calculation of the terms of these sequences requires the determination of the smallest... a rather indirect proof, which, in some sense, is unnatural; but, on the other hand, as I shall indicate, it leads to the most important developments Euler showed that there must exist infinitely many primes because a certain expression formed with all the primes is infinite If p is any prime, then 1/p < 1; hence, the sum of the geometric series is ∞ 1 1 = 1 − (1/p) pk k=0 Similarly, if q is another . alt="" The Little Book of Bigger Primes Second Edition