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- 1. Martin Aigner Günter M. ZieglerProofs from THE BOOK Fourth Edition
- 2. Martin AignerGünter M. ZieglerProofs fromTHE BOOKFourth EditionIncluding Illustrations by Karl H. Hofmann123
- 3. Prof. Dr. Martin Aigner Prof. Günter M. ZieglerFB Mathematik und Informatik Institut für Mathematik, MA 6-2Freie Universität Berlin Technische Universität BerlinArnimallee 3 Straße des 17. Juni 13614195 Berlin 10623 BerlinDeutschland Deutschlandaigner@math.fu-berlin.de ziegler@math.tu-berlin.deISBN 978-3-642-00855-9 e-ISBN 978-3-642-00856-6DOI 10.1007/978-3-642-00856-6Springer Heidelberg Dordrecht London New York c Springer-Verlag Berlin Heidelberg 2010This work is subject to copyright. All rights are reserved, whether the whole or part of thematerial is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilm or in any other way, and storage in databanks. Duplication of this publication or parts thereof is permitted only under the provisions ofthe German Copyright Law of September 9, 1965, in its current version, and permission for usemust always be obtained from Springer. Violations are liable to prosecution under the GermanCopyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.Cover design: deblik, BerlinPrinted on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)
- 4. PrefacePaul Erd˝ s liked to talk about The Book, in which God maintains the perfect oproofs for mathematical theorems, following the dictum of G. H. Hardy thatthere is no permanent place for ugly mathematics. Erd˝ s also said that you oneed not believe in God but, as a mathematician, you should believe inThe Book. A few years ago, we suggested to him to write up a ﬁrst (andvery modest) approximation to The Book. He was enthusiastic about theidea and, characteristically, went to work immediately, ﬁlling page afterpage with his suggestions. Our book was supposed to appear in March1998 as a present to Erd˝ s’ 85th birthday. With Paul’s unfortunate death oin the summer of 1996, he is not listed as a co-author. Instead this book isdedicated to his memory. Paul Erd˝ s oWe have no deﬁnition or characterization of what constitutes a proof fromThe Book: all we offer here is the examples that we have selected, hop-ing that our readers will share our enthusiasm about brilliant ideas, cleverinsights and wonderful observations. We also hope that our readers willenjoy this despite the imperfections of our exposition. The selection is to agreat extent inﬂuenced by Paul Erd˝ s himself. A large number of the topics owere suggested by him, and many of the proofs trace directly back to him,or were initiated by his supreme insight in asking the right question or inmaking the right conjecture. So to a large extent this book reﬂects the viewsof Paul Erd˝ s as to what should be considered a proof from The Book. o “The Book”A limiting factor for our selection of topics was that everything in this bookis supposed to be accessible to readers whose backgrounds include onlya modest amount of technique from undergraduate mathematics. A littlelinear algebra, some basic analysis and number theory, and a healthy dollopof elementary concepts and reasonings from discrete mathematics shouldbe sufﬁcient to understand and enjoy everything in this book.We are extremely grateful to the many people who helped and supportedus with this project — among them the students of a seminar where wediscussed a preliminary version, to Benno Artmann, Stephan Brandt, StefanFelsner, Eli Goodman, Torsten Heldmann, and Hans Mielke. We thankMargrit Barrett, Christian Bressler, Ewgenij Gawrilow, Michael Joswig,Elke Pose, and Jörg Rambau for their technical help in composing thisbook. We are in great debt to Tom Trotter who read the manuscript fromﬁrst to last page, to Karl H. Hofmann for his wonderful drawings, andmost of all to the late great Paul Erd˝ s himself. oBerlin, March 1998 Martin Aigner · Günter M. Ziegler
- 5. VI Preface to the Fourth Edition When we started this project almost ﬁfteen years ago we could not possibly imagine what a wonderful and lasting response our book about The Book would have, with all the warm letters, interesting comments, new editions, and as of now thirteen translations. It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements, partly suggested by our readers, the present fourth edition contains ﬁve new chapters: Two classics, the law of quadratic reciprocity and the fundamental theorem of algebra, two chapters on tiling problems and their intriguing solutions, and a highlight in graph theory, the chromatic number of Kneser graphs. We thank everyone who helped and encouraged us over all these years: For the second edition this included Stephan Brandt, Christian Elsholtz, Jürgen Elstrodt, Daniel Grieser, Roger Heath-Brown, Lee L. Keener, Christian Lebœuf, Hanfried Lenz, Nicolas Puech, John Scholes, Bernulf Weißbach, and many others. The third edition beneﬁtted especially from input by David Bevan, Anders Björner, Dietrich Braess, John Cosgrave, Hubert Kalf, Günter Pickert, Alistair Sinclair, and Herb Wilf. For the present edi- tion, we are particularly grateful to contributions by France Dacar, Oliver Deiser, Anton Dochtermann, Michael Harbeck, Stefan Hougardy, Hendrik W. Lenstra, Günter Rote, Moritz Schmitt, and Carsten Schultz. Moreover, we thank Ruth Allewelt at Springer in Heidelberg as well as Christoph Eyrich, Torsten Heldmann, and Elke Pose in Berlin for their help and sup- port throughout these years. And ﬁnally, this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch, and the superb new drawings provided for each edition by Karl H. Hofmann. Berlin, July 2009 Martin Aigner · Günter M. Ziegler
- 6. Table of ContentsNumber Theory 1 1. Six proofs of the inﬁnity of primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Bertrand’s postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. Binomial coefﬁcients are (almost) never powers . . . . . . . . . . . . . . . . . 13 4. Representing numbers as sums of two squares . . . . . . . . . . . . . . . . . . . 17 5. The law of quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6. Every ﬁnite division ring is a ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7. Some irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8. Three times π 2 /6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Geometry 51 9. Hilbert’s third problem: decomposing polyhedra . . . . . . . . . . . . . . . . . 5310. Lines in the plane and decompositions of graphs . . . . . . . . . . . . . . . . . 6311. The slope problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6912. Three applications of Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 7513. Cauchy’s rigidity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114. Touching simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515. Every large point set has an obtuse angle . . . . . . . . . . . . . . . . . . . . . . . 8916. Borsuk’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Analysis 10117. Sets, functions, and the continuum hypothesis . . . . . . . . . . . . . . . . . . 10318. In praise of inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11919. The fundamental theorem of algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 12720. One square and an odd number of triangles . . . . . . . . . . . . . . . . . . . . 131
- 7. VIII Table of Contents 21. A theorem of Pólya on polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 22. On a lemma of Littlewood and Offord . . . . . . . . . . . . . . . . . . . . . . . . . 145 23. Cotangent and the Herglotz trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 24. Buffon’s needle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Combinatorics 159 25. Pigeon-hole and double counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 26. Tiling rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 27. Three famous theorems on ﬁnite sets . . . . . . . . . . . . . . . . . . . . . . . . . . 179 28. Shufﬂing cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 29. Lattice paths and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 30. Cayley’s formula for the number of trees . . . . . . . . . . . . . . . . . . . . . . 201 31. Identities versus bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 32. Completing Latin squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Graph Theory 219 33. The Dinitz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221 34. Five-coloring plane graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 35. How to guard a museum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 36. Turán’s graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 37. Communicating without errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 38. The chromatic number of Kneser graphs . . . . . . . . . . . . . . . . . . . . . . . 251 39. Of friends and politicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 40. Probability makes counting (sometimes) easy . . . . . . . . . . . . . . . . . . 261 About the Illustrations 270 Index 271
- 8. Number Theory 1 Six proofs of the inﬁnity of primes 3 2 Bertrand’s postulate 7 3 Binomial coefﬁcients are (almost) never powers 13 4 Representing numbers as sums of two squares 17 5 The law of quadratic reciprocity 23 6 Every ﬁnite division ring is a ﬁeld 31 7 Some irrational numbers 35 8 Three times π 2 /6 43 “Irrationality and π”
- 9. Six proofs Chapter 1of the inﬁnity of primesIt is only natural that we start these notes with probably the oldest BookProof, usually attributed to Euclid (Elements IX, 20). It shows that thesequence of primes does not end. Euclid’s Proof. For any ﬁnite set {p1 , . . . , pr } of primes, considerthe number n = p1 p2 · · · pr + 1. This n has a prime divisor p. But p isnot one of the pi : otherwise p would be a divisor of n and of the productp1 p2 · · · pr , and thus also of the difference n − p1 p2 · · · pr = 1, which isimpossible. So a ﬁnite set {p1 , . . . , pr } cannot be the collection of all primenumbers.Before we continue let us ﬁx some notation. N = {1, 2, 3, . . .} is the setof natural numbers, Z = {. . . , −2, −1, 0, 1, 2, . . .} the set of integers, andP = {2, 3, 5, 7, . . .} the set of primes.In the following, we will exhibit various other proofs (out of a much longerlist) which we hope the reader will like as much as we do. Although theyuse different view-points, the following basic idea is common to all of them:The natural numbers grow beyond all bounds, and every natural numbern ≥ 2 has a prime divisor. These two facts taken together force P to beinﬁnite. The next proof is due to Christian Goldbach (from a letter to Leon-hard Euler 1730), the third proof is apparently folklore, the fourth one isby Euler himself, the ﬁfth proof was proposed by Harry Fürstenberg, whilethe last proof is due to Paul Erd˝ s. o n F0 = 3 Second Proof. Let us ﬁrst look at the Fermat numbers Fn = 22 +1 for F1 = 5n = 0, 1, 2, . . .. We will show that any two Fermat numbers are relatively F2 = 17prime; hence there must be inﬁnitely many primes. To this end, we verify F3 = 257the recursion n−1 F4 = 65537 Fk = Fn − 2 (n ≥ 1), F5 = 641 · 6700417 k=0 The ﬁrst few Fermat numbersfrom which our assertion follows immediately. Indeed, if m is a divisor of,say, Fk and Fn (k < n), then m divides 2, and hence m = 1 or 2. Butm = 2 is impossible since all Fermat numbers are odd.To prove the recursion we use induction on n. For n = 1 we have F0 = 3and F1 − 2 = 3. With induction we now conclude n n−1 Fk = Fk Fn = (Fn − 2)Fn = k=0 k=0 n n n+1 = (22 − 1)(22 + 1) = 22 − 1 = Fn+1 − 2.
- 10. 4 Six proofs of the inﬁnity of primes Third Proof. Suppose P is ﬁnite and p is the largest prime. We consider Lagrange’s Theorem the so-called Mersenne number 2p − 1 and show that any prime factor q If G is a ﬁnite (multiplicative) group of 2p − 1 is bigger than p, which will yield the desired conclusion. Let q be and U is a subgroup, then |U | a prime dividing 2p − 1, so we have 2p ≡ 1 (mod q). Since p is prime, this divides |G|. means that the element 2 has order p in the multiplicative group Zq {0} of Proof. Consider the binary rela- the ﬁeld Zq . This group has q − 1 elements. By Lagrange’s theorem (see tion the box) we know that the order of every element divides the size of the a ∼ b : ⇐⇒ ba−1 ∈ U. group, that is, we have p | q − 1, and hence p < q. It follows from the group axioms Now let us look at a proof that uses elementary calculus. that ∼ is an equivalence relation. Fourth Proof. Let π(x) := #{p ≤ x : p ∈ P} be the number of primes The equivalence class containing an that are less than or equal to the real number x. We number the primes element a is precisely the coset P = {p1 , p2 , p3 , . . .} in increasing order. Consider the natural logarithm x U a = {xa : x ∈ U }. log x, deﬁned as log x = 1 1 dt. t Now we compare the area below the graph of f (t) = 1 with an upper step t Since clearly |U a| = |U |, we ﬁnd function. (See also the appendix on page 10 for this method.) Thus for that G decomposes into equivalence n ≤ x < n + 1 we have classes, all of size |U |, and hence that |U | divides |G|. 1 1 1 1 log x ≤ 1+ + + ...+ + 2 3 n−1 n In the special case when U is a cyclic subgroup {a, a2 , . . . , am } we ﬁnd 1 ≤ , where the sum extends over all m ∈ N which have that m (the smallest positive inte- m only prime divisors p ≤ x. ger such that am = 1, called the order of a) divides the size |G| of Since every such m can be written in a unique way as a product of the form the group. pkp , we see that the last sum is equal to p≤x 1 . pk p∈P k≥0 p≤x1 1 The inner sum is a geometric series with ratio p , hence π(x) 1 p pk log x ≤ 1 = = . 1− p p−1 pk − 1 p∈P p∈P k=1 p≤x p≤x 1 2 n n+1 Steps above the function f (t) = 1 t Now clearly pk ≥ k + 1, and thus pk 1 1 k+1 = 1+ ≤ 1+ = , pk − 1 pk − 1 k k and therefore π(x) k+1 log x ≤ = π(x) + 1. k k=1 Everybody knows that log x is not bounded, so we conclude that π(x) is unbounded as well, and so there are inﬁnitely many primes.
- 11. Six proofs of the inﬁnity of primes 5 Fifth Proof. After analysis it’s topology now! Consider the followingcurious topology on the set Z of integers. For a, b ∈ Z, b > 0, we set Na,b = {a + nb : n ∈ Z}.Each set Na,b is a two-way inﬁnite arithmetic progression. Now call a setO ⊆ Z open if either O is empty, or if to every a ∈ O there exists someb > 0 with Na,b ⊆ O. Clearly, the union of open sets is open again. IfO1 , O2 are open, and a ∈ O1 ∩ O2 with Na,b1 ⊆ O1 and Na,b2 ⊆ O2 ,then a ∈ Na,b1 b2 ⊆ O1 ∩ O2 . So we conclude that any ﬁnite intersectionof open sets is again open. So, this family of open sets induces a bona ﬁdetopology on Z.Let us note two facts:(A) Any nonempty open set is inﬁnite.(B) Any set Na,b is closed as well.Indeed, the ﬁrst fact follows from the deﬁnition. For the second we observe b−1 Na,b = Z Na+i,b , i=1which proves that Na,b is the complement of an open set and hence closed.So far the primes have not yet entered the picture — but here they come.Since any number n = 1, −1 has a prime divisor p, and hence is containedin N0,p , we conclude Z {1, −1} = N0,p . “Pitching ﬂat rocks, inﬁnitely” p∈PNow if P were ﬁnite, then p∈P N0,p would be a ﬁnite union of closed sets(by (B)), and hence closed. Consequently, {1, −1} would be an open set,in violation of (A). Sixth Proof. Our ﬁnal proof goes a considerable step further anddemonstrates not only that there are inﬁnitely many primes, but also thatthe series p∈P 1 diverges. The ﬁrst proof of this important result was pgiven by Euler (and is interesting in its own right), but our proof, devisedby Erd˝ s, is of compelling beauty. oLet p1 , p2 , p3 , . . . be the sequence of primes in increasing order, and 1assume that p∈P p converges. Then there must be a natural number k 1 1such that i≥k+1 pi < 2 . Let us call p1 , . . . , pk the small primes, andpk+1 , pk+2 , . . . the big primes. For an arbitrary natural number N we there-fore ﬁnd N N < . (1) pi 2 i≥k+1
- 12. 6 Six proofs of the inﬁnity of primes Let Nb be the number of positive integers n ≤ N which are divisible by at least one big prime, and Ns the number of positive integers n ≤ N which have only small prime divisors. We are going to show that for a suitable N Nb + Ns < N, which will be our desired contradiction, since by deﬁnition Nb + Ns would have to be equal to N . N To estimate Nb note that ⌊ pi ⌋ counts the positive integers n ≤ N which are multiples of pi . Hence by (1) we obtain N N Nb ≤ < . (2) pi 2 i≥k+1 Let us now look at Ns . We write every n ≤ N which has only small prime divisors in the form n = an b2 , where an is the square-free part. Every an n is thus a product of different small primes, and we conclude that there are √ √ precisely 2k different square-free parts. Furthermore, as bn ≤ n ≤ N , √ we ﬁnd that there are at most N different square parts, and so √ Ns ≤ 2 k N . √ Since (2) holds for any N , it remains to ﬁnd a number N with 2k N ≤ N √ 2 or 2k+1 ≤ N , and for this N = 22k+2 will do. References [1] B. A RTMANN : Euclid — The Creation of Mathematics, Springer-Verlag, New York 1999. P1 ˝ [2] P. E RD OS : Über die Reihe , Mathematica, Zutphen B 7 (1938), 1-2. p [3] L. E ULER : Introductio in Analysin Inﬁnitorum, Tomus Primus, Lausanne 1748; Opera Omnia, Ser. 1, Vol. 8. [4] H. F ÜRSTENBERG : On the inﬁnitude of primes, Amer. Math. Monthly 62 (1955), 353.
- 13. Bertrand’s postulate Chapter 2We have seen that the sequence of prime numbers 2, 3, 5, 7, . . . is inﬁnite.To see that the size of its gaps is not bounded, let N := 2 · 3 · 5 · · · p denotethe product of all prime numbers that are smaller than k + 2, and note thatnone of the k numbers N + 2, N + 3, N + 4, . . . , N + k, N + (k + 1)is prime, since for 2 ≤ i ≤ k + 1 we know that i has a prime factor that issmaller than k + 2, and this factor also divides N , and hence also N + i.With this recipe, we ﬁnd, for example, for k = 10 that none of the tennumbers 2312, 2313, 2314, . . ., 2321is prime.But there are also upper bounds for the gaps in the sequence of prime num-bers. A famous bound states that “the gap to the next prime cannot be largerthan the number we start our search at.” This is known as Bertrand’s pos-tulate, since it was conjectured and veriﬁed empirically for n < 3 000 000by Joseph Bertrand. It was ﬁrst proved for all n by Pafnuty Chebyshev in Joseph Bertrand1850. A much simpler proof was given by the Indian genius Ramanujan.Our Book Proof is by Paul Erd˝ s: it is taken from Erd˝ s’ ﬁrst published o opaper, which appeared in 1932, when Erd˝ s was 19. o Bertrand’s postulate. For every n ≥ 1, there is some prime number p with n < p ≤ 2n. Proof. We will estimate the size of the binomial coefﬁcient 2n care- nfully enough to see that if it didn’t have any prime factors in the rangen < p ≤ 2n, then it would be “too small.” Our argument is in ﬁve steps.(1) We ﬁrst prove Bertrand’s postulate for n < 4000. For this one does notneed to check 4000 cases: it sufﬁces (this is “Landau’s trick”) to check that 2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001is a sequence of prime numbers, where each is smaller than twice the previ-ous one. Hence every interval {y : n < y ≤ 2n}, with n ≤ 4000, containsone of these 14 primes.
- 14. 8 Bertrand’s postulate (2) Next we prove that p ≤ 4x−1 for all real x ≥ 2, (1) p≤x where our notation — here and in the following — is meant to imply that the product is taken over all prime numbers p ≤ x. The proof that we present for this fact uses induction on the number of these primes. It is not from Erd˝ s’ original paper, but it is also due to Erd˝ s (see the margin), o o and it is a true Book Proof. First we note that if q is the largest prime with q ≤ x, then p = p and 4q−1 ≤ 4x−1 . p≤x p≤q Thus it sufﬁces to check (1) for the case where x = q is a prime number. For q = 2 we get “2 ≤ 4,” so we proceed to consider odd primes q = 2m + 1. (Here we may assume, by induction, that (1) is valid for all integers x in the set {2, 3, . . . , 2m}.) For q = 2m + 1 we split the product and compute 2m + 1 p = p· p ≤ 4m ≤ 4m 22m = 42m. m p≤2m+1 p≤m+1 m+1<p≤2m+1 All the pieces of this “one-line computation” are easy to see. In fact, p ≤ 4m p≤m+1 holds by induction. The inequality 2m + 1 p ≤ m m+1<p≤2m+1 (2m+1)! follows from the observation that 2m+1 = m!(m+1)! is an integer, where m the primes that we consider all are factors of the numerator (2m + 1)!, but not of the denominator m!(m + 1)!. Finally Legendre’s theorem 2m + 1 ≤ 22m m The number n! contains the prime holds since factor p exactly 2m + 1 2m + 1 Xj n k and m m+1 pk k≥1 are two (equal!) summands that appear in times. 2m+1 2m + 1 Proof. Exactly n of the factors ¨ ˝ = 22m+1 . p k of n! = 1 · 2 · 3 · · · n are divisible by k=0 p, which accounts for n p-factors. ¨ ˝ p 2n (2n)! ¨n˝ Next, p2 of the factors of n! are (3) From Legendre’s theorem (see the box) we get that n = n!n! con- even divisible by p2 , which accounts tains the prime factor p exactly ¨n˝ for the next p2 prime factors p 2n n −2 k of n!, etc. pk p k≥1
- 15. Bertrand’s postulate 9times. Here each summand is at most 1, since it satisﬁes 2n n 2n n −2 k < −2 −1 = 2, pk p pk pkand it is an integer. Furthermore the summands vanish whenever pk > 2n.Thus 2n contains p exactly n 2n n k −2 k ≤ max{r : pr ≤ 2n} p p k≥1times. Hence the largest power of p that divides 2n is not larger than 2n. √ nIn particular, primes p > 2n appear at most once in 2n . nFurthermore — and this, according to Erd˝ s, is the key fact for his proof o Examples such as— primes p that satisfy 2 n < p ≤ n do not divide 2n at all! Indeed, `26´ 3 n 13 = 23 · 52 · 7 · 17 · 19 · 233p > 2n implies (for n ≥ 3, and hence p ≥ 3) that p and 2p are the only `28´ 14 = 23 · 33 · 52 · 17 · 19 · 23multiples of p that appear as factors in the numerator of (2n)! , while we get `30´ n!n! 15 = 24 · 32 · 5 · 17 · 19 · 23 · 29two p-factors in the denominator. illustrate that “very small” prime factors √(4) Now we are ready to estimate 2n . For n ≥ 3, using an estimate from p < 2n can appear as higher powers n `2n´ √page 12 for the lower bound, we get in n , “small” primes with 2n < p ≤ 2 n appear at most once, while 3 4n 2n 2 factors in the gap with 3 n < p ≤ n ≤ ≤ 2n · p · p 2n n √ √ don’t appear at all. p≤ 2n 2n<p≤ 2 n n<p≤2n 3 √ √and thus, since there are not more than 2n primes p ≤ 2n, √ 4n ≤ (2n)1+ 2n · p · p for n ≥ 3. (2) √ 2n<p≤ 2 n n<p≤2n 3(5) Assume now that there is no prime p with n < p ≤ 2n, so the secondproduct in (2) is 1. Substituting (1) into (2) we get √ 2 4n ≤ (2n)1+ 2n 43nor 1 √ 4 3 n ≤ (2n)1+ 2n , (3)which is false for n large enough! In fact, using a + 1 < 2 (which holds afor all a ≥ 2, by induction) we get √ 6 6 √ 6 6 √ 6 √ 6 2n = 2n < 2n + 1 < 26 2n ≤ 26 2n , (4) √and thus for n ≥ 50 (and hence 18 < 2 2n) we obtain from (3) and (4) √ √ 6 √ √ 6 √ 2/3 22n ≤ (2n)3 1+ 2n <2 2n 18+18 2n < 220 2n 2n = 220(2n) .This implies (2n)1/3 < 20, and thus n < 4000.
- 16. 10 Bertrand’s postulate One can extract even more from this type of estimates: From (2) one can derive with the same methods that 1 p ≥ 2 30 n for n ≥ 4000, n<p≤2n and thus that there are at least 1 1 n log2n 2 30 n = 30 log2 n + 1 primes in the range between n and 2n. This is not that bad an estimate: the “true” number of primes in this range is roughly n/ log n. This follows from the “prime number theorem,” which says that the limit #{p ≤ n : p is prime} lim n→∞ n/ log n exists, and equals 1. This famous result was ﬁrst proved by Hadamard and de la Vallée-Poussin in 1896; Selberg and Erd˝ s found an elementary proof o (without complex analysis tools, but still long and involved) in 1948. On the prime number theorem itself the ﬁnal word, it seems, is still not in: for example a proof of the Riemann hypothesis (see page 49), one of the major unsolved open problems in mathematics, would also give a substan- tial improvement for the estimates of the prime number theorem. But also for Bertrand’s postulate, one could expect dramatic improvements. In fact, the following is a famous unsolved problem: Is there always a prime between n2 and (n + 1)2 ? For additional information see [3, p. 19] and [4, pp. 248, 257]. Appendix: Some estimates Estimating via integrals There is a very simple-but-effective method of estimating sums by integrals (as already encountered on page 4). For estimating the harmonic numbers n 1 Hn = k k=1 we draw the ﬁgure in the margin and derive from it n n 1 1 Hn − 1 = < dt = log n1 k 1 t k=2 1 by comparing the area below the graph of f (t) = t (1 ≤ t ≤ n) with the area of the dark shaded rectangles, and n−1 n 1 1 1 Hn − = > dt = log n n k 1 t 1 2 n k=1
- 17. Bertrand’s postulate 11by comparing with the area of the large rectangles (including the lightlyshaded parts). Taken together, this yields 1 log n + < Hn < log n + 1. nIn particular, lim Hn → ∞, and the order of growth of Hn is given by n→∞ Hnlim = 1. But much better estimates are known (see [2]), such asn→∞ log n ` 1´ Here O n6 denotes a function f (n) 1 1 1 1 1 such that f (n) ≤ c n6 holds for some Hn = log n + γ + − 2 + +O , 2n 12n 120n4 n6 constant c.where γ ≈ 0.5772 is “Euler’s constant.”Estimating factorials — Stirling’s formulaThe same method applied to n log(n!) = log 2 + log 3 + . . . + log n = log k k=2yields n log((n − 1)!) < log t dt < log(n!), 1where the integral is easily computed: n n log t dt = t log t − t = n log n − n + 1. 1 1Thus we get a lower estimate on n! n n n! > en log n−n+1 = e eand at the same time an upper estimate n n n! = n (n − 1)! < nen log n−n+1 = en . eHere a more careful analysis is needed to get the asymptotics of n!, as givenby Stirling’s formula Here f (n) ∼ g(n) means that √ n n n! ∼ 2πn . f (n) e lim = 1. n→∞ g(n)And again there are more precise versions available, such as √ n n 1 1 139 1 n! = 2πn 1+ + − +O . e 12n 288n2 5140n3 n4Estimating binomial coefﬁcients nJust from the deﬁnition of the binomial coefﬁcients k as the number of nk-subsets of an n-set, we know that the sequence 0 , n , . . . , n of 1 nbinomial coefﬁcients
- 18. 12 Bertrand’s postulate n n • sums to k = 2n k=0 1 n n • is symmetric: k = n−k . 1 1 1 2 1 1 3 3 1 From the functional equation n = n−k+1 k−1 one easily ﬁnds that for k k n 1 6 4 1 4 n every n the binomial coefﬁcients k form a sequence that is symmetric 1 5 10 10 5 1 1 6 15 20 15 6 1 and unimodal: it increases towards the middle, so that the middle binomial1 7 21 35 35 21 7 1 coefﬁcients are the largest ones in the sequence:Pascal’s triangle n n n n n n 1= 0 < 1 < ··· < ⌊n/2⌋ = ⌈n/2⌉ > ··· > n−1 > n = 1. Here ⌊x⌋ resp. ⌈x⌉ denotes the number x rounded down resp. rounded up to the nearest integer. From the asymptotic formulas for the factorials mentioned above one can obtain very precise estimates for the sizes of binomial coefﬁcients. How- ever, we will only need very weak and simple estimates in this book, such as the following: n ≤ 2n for all k, while for n ≥ 2 we have k n 2n ≥ , ⌊n/2⌋ n with equality only for n = 2. In particular, for n ≥ 1, 2n 4n ≥ . n 2n n This holds since ⌊n/2⌋ , a middle binomial coefﬁcient, is the largest entry n in the sequence 0 + n , n , n , . . . , n−1 , whose sum is 2n , and whose n 1 2 n n average is thus 2n . On the other hand, we note the upper bound for binomial coefﬁcients n n(n − 1) · · · (n − k + 1) nk nk = ≤ ≤ k−1 , k k! k! 2 which is a reasonably good estimate for the “small” binomial coefﬁcients at the tails of the sequence, when n is large (compared to k). References ˝ [1] P. E RD OS : Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5 (1930-32), 194-198. [2] R. L. G RAHAM , D. E. K NUTH & O. PATASHNIK : Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Reading MA 1989. [3] G. H. H ARDY & E. M. W RIGHT: An Introduction to the Theory of Numbers, ﬁfth edition, Oxford University Press 1979. [4] P. R IBENBOIM : The New Book of Prime Number Records, Springer-Verlag, New York 1989.
- 19. Binomial coefﬁcients Chapter 3are (almost) never powersThere is an epilogue to Bertrand’s postulate which leads to a beautiful re-sult on binomial coefﬁcients. In 1892 Sylvester strengthened Bertrand’spostulate in the following way: If n ≥ 2k, then at least one of the numbers n, n − 1, . . . , n − k + 1 has a prime divisor p greater than k.Note that for n = 2k we obtain precisely Bertrand’s postulate. In 1934,Erd˝ s gave a short and elementary Book Proof of Sylvester’s result, running oalong the lines of his proof of Bertrand’s postulate. There is an equivalentway of stating Sylvester’s theorem: The binomial coefﬁcient n n(n − 1) · · · (n − k + 1) = (n ≥ 2k) k k! always has a prime factor p > k.With this observation in mind, we turn to another one of Erd˝ s’ jewels. oWhen is n equal to a power mℓ ? It is easy to see that there are inﬁnitely kmany solutions for k = ℓ = 2, that is, of the equation n = m2 . Indeed, 2 (2n−1)2if n is a square, then so is 2 2 . To see this, set n(n − 1) = 2m2 .It follows that (2n − 1)2 ((2n − 1)2 − 1) = (2n − 1)2 4n(n − 1) = 2(2m(2n − 1))2 ,and hence (2n − 1)2 = (2m(2n − 1))2 . 2Beginning with 9 = 62 we thus obtain inﬁnitely many solutions — the 2next one is 289 = 2042. However, this does not yield all solutions. For 2example, 50 = 352 starts another series, as does 1682 = 11892. For 2 2 `50´k = 3 it is known that n = m2 has the unique solution n = 50, m = 140. 3 = 1402 3But now we are at the end of the line. For k ≥ 4 and any ℓ ≥ 2 no solutions is the only solution for k = 3, ℓ = 2exist, and this is what Erd˝ s proved by an ingenious argument. o Theorem. The equation n = mℓ has no integer solutions with k ℓ ≥ 2 and 4 ≤ k ≤ n − 4.
- 20. 14 Binomial coefﬁcients are (almost) never powers Proof. Note ﬁrst that we may assume n ≥ 2k because of n = n−k . k n n Suppose the theorem is false, and that k = m . The proof, by contra- ℓ diction, proceeds in the following four steps. (1) By Sylvester’s theorem, there is a prime factor p of n greater than k, k hence pℓ divides n(n − 1) · · · (n − k + 1). Clearly, only one of the factors n − i can be a multiple of p (because of p > k), and we conclude pℓ | n − i, and therefore n ≥ pℓ > k ℓ ≥ k 2 . (2) Consider any factor n − j of the numerator and write it in the form n − j = aj mℓ , where aj is not divisible by any nontrivial ℓ-th power. We j note by (1) that aj has only prime divisors less than or equal to k. We want to show next that ai = aj for i = j. Assume to the contrary that ai = aj for some i < j. Then mi ≥ mj + 1 and k > (n − i) − (n − j) = aj (mℓ − mℓ ) ≥ aj ((mj + 1)ℓ − mℓ ) i j j > aj ℓmℓ−1 ≥ ℓ(aj mℓ )1/2 ≥ ℓ(n − k + 1)1/2 j j ≥ ℓ( n + 1)1/2 > n1/2 , 2 which contradicts n > k 2 from above. (3) Next we prove that the ai ’s are the integers 1, 2, . . . , k in some order. (According to Erd˝ s, this is the crux of the proof.) Since we already know o that they are all distinct, it sufﬁces to prove that a0 a1 · · · ak−1 divides k!. n Substituting n − j = aj mℓ into the equation j k = mℓ , we obtain a0 a1 · · · ak−1 (m0 m1 · · · mk−1 )ℓ = k!mℓ . Cancelling the common factors of m0 · · · mk−1 and m yields a0 a1 · · · ak−1 uℓ = k!v ℓ with gcd(u, v) = 1. It remains to show that v = 1. If not, then v con- tains a prime divisor p. Since gcd(u, v) = 1, p must be a prime divisor of a0 a1 · · · ak−1 and hence is less than or equal to k. By the theorem of k Legendre (see page 8) we know that k! contains p to the power i≥1 ⌊ pi ⌋. We now estimate the exponent of p in n(n − 1) · · · (n − k + 1). Let i be a positive integer, and let b1 < b2 < · · · < bs be the multiples of pi among n, n − 1, . . . , n − k + 1. Then bs = b1 + (s − 1)pi and hence (s − 1)pi = bs − b1 ≤ n − (n − k + 1) = k − 1, which implies k−1 k s ≤ +1 ≤ + 1. pi pi
- 21. Binomial coefﬁcients are (almost) never powers 15So for each i the number of multiples of pi among n, . . . , n−k+1, and khence among the aj ’s, is bounded by ⌊ pi ⌋ + 1. This implies that the expo-nent of p in a0 a1 · · · ak−1 is at most ℓ−1 k +1 i=1 piwith the reasoning that we used for Legendre’s theorem in Chapter 2. Theonly difference is that this time the sum stops at i = ℓ − 1, since the aj ’scontain no ℓ-th powers.Taking both counts together, we ﬁnd that the exponent of p in v ℓ is at most ℓ−1 k k +1 − ≤ ℓ − 1, i=1 pi pi i≥1and we have our desired contradiction, since v ℓ is an ℓ-th power.This sufﬁces already to settle the case ℓ = 2. Indeed, since k ≥ 4 one of We see that our analysis so far agreesthe ai ’s must be equal to 4, but the ai ’s contain no squares. So let us now with 50 = 1402 , as ` ´ 3assume that ℓ ≥ 3. 50 = 2 · 52(4) Since k ≥ 4, we must have ai1 = 1, ai2 = 2, ai3 = 4 for some i1 , i2 , i3 , 49 = 1 · 72 48 = 3 · 42that is, and 5 · 7 · 4 = 140. n − i1 = mℓ , n − i2 = 2mℓ , n − i3 = 4mℓ . 1 2 3We claim that (n − i2 )2 = (n − i1 )(n − i3 ). If not, put b = n − i2 andn − i1 = b − x, n − i3 = b + y, where 0 < |x|, |y| < k. Hence b2 = (b − x)(b + y) or (y − x)b = xy,where x = y is plainly impossible. Now we have by part (1) |xy| = b|y − x| ≥ b > n − k > (k − 1)2 ≥ |xy|,which is absurd.So we have m2 = m1 m3 , where we assume m2 > m1 m3 (the other case 2 2being analogous), and proceed to our last chains of inequalities. We obtain 2(k − 1)n > n2 − (n − k + 1)2 > (n − i2 )2 − (n − i1 )(n − i3 ) = 4[m2ℓ − (m1 m3 )ℓ ] ≥ 4[(m1 m3 + 1)ℓ − (m1 m3 )ℓ ] 2 ≥ 4ℓmℓ−1mℓ−1 . 1 3Since ℓ ≥ 3 and n > k ℓ ≥ k 3 > 6k, this yields 2(k − 1)nm1 m3 > 4ℓmℓ mℓ = ℓ(n − i1 )(n − i3 ) 1 3 n > ℓ(n − k + 1)2 > 3(n − )2 > 2n2 . 6
- 22. 16 Binomial coefﬁcients are (almost) never powers Now since mi ≤ n1/ℓ ≤ n1/3 we ﬁnally obtain kn2/3 ≥ km1 m3 > (k − 1)m1 m3 > n, or k 3 > n. With this contradiction, the proof is complete. References ˝ [1] P. E RD OS : A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), 282-288. ˝ [2] P. E RD OS : On a diophantine equation, J. London Math. Soc. 26 (1951), 176-178. [3] J. J. S YLVESTER : On arithmetical series, Messenger of Math. 21 (1892), 1-19, 87-120; Collected Mathematical Papers Vol. 4, 1912, 687-731.
- 23. Representing numbers Chapter 4as sums of two squares 1 = 12 + 02 Which numbers can be written as sums of two squares? 2 = 12 + 12 3 = 4 = 22 + 02This question is as old as number theory, and its solution is a classic in the 5 = 22 + 12ﬁeld. The “hard” part of the solution is to see that every prime number of 6 =the form 4m + 1 is a sum of two squares. G. H. Hardy writes that this 7 =two square theorem of Fermat “is ranked, very justly, as one of the ﬁnest in 8 = 22 + 22arithmetic.” Nevertheless, one of our Book Proofs below is quite recent. 9 = 32 +Let’s start with some “warm-ups.” First, we need to distinguish between 10 = 32 +the prime p = 2, the primes of the form p = 4m + 1, and the primes of 11 = . .the form p = 4m + 3. Every prime number belongs to exactly one of these .three classes. At this point we may note (using a method “à la Euclid”) thatthere are inﬁnitely many primes of the form 4m + 3. In fact, if there wereonly ﬁnitely many, then we could take pk to be the largest prime of thisform. Setting Nk := 22 · 3 · 5 · · · pk − 1(where p1 = 2, p2 = 3, p3 = 5, . . . denotes the sequence of all primes), weﬁnd that Nk is congruent to 3 (mod 4), so it must have a prime factor of theform 4m + 3, and this prime factor is larger than pk — contradiction.Our ﬁrst lemma characterizes the primes for which −1 is a square in the Pierre de Fermatﬁeld Zp (which is reviewed in the box on the next page). It will also giveus a quick way to derive that there are inﬁnitely many primes of the form4m + 1.Lemma 1. For primes p = 4m + 1 the equation s2 ≡ −1 (mod p) has twosolutions s ∈ {1, 2, . . ., p−1}, for p = 2 there is one such solution, whilefor primes of the form p = 4m + 3 there is no solution. Proof. For p = 2 take s = 1. For odd p, we construct the equivalencerelation on {1, 2, . . . , p − 1} that is generated by identifying every elementwith its additive inverse and with its multiplicative inverse in Zp . Thus the“general” equivalence classes will contain four elements {x, −x, x, −x}since such a 4-element set contains both inverses for all its elements. How-ever, there are smaller equivalence classes if some of the four numbers arenot distinct:
- 24. 18 Representing numbers as sums of two squares • x ≡ −x is impossible for odd p. • x ≡ x is equivalent to x2 ≡ 1. This has two solutions, namely x = 1 and x = p − 1, leading to the equivalence class {1, p − 1} of size 2. • x ≡ −x is equivalent to x2 ≡ −1. This equation may have no solution or two distinct solutions x0 , p − x0 : in this case the equivalence class is {x0 , p − x0 }.For p = 11 the partition is The set {1, 2, . . . , p − 1} has p − 1 elements, and we have partitioned it into{1, 10}, {2, 9, 6, 5}, {3, 8, 4, 7}; quadruples (equivalence classes of size 4), plus one or two pairs (equiva-for p = 13 it is lence classes of size 2). For p − 1 = 4m + 2 we ﬁnd that there is only the{1, 12}, {2, 11, 7, 6}, {3, 10, 9, 4}, one pair {1, p− 1}, the rest is quadruples, and thus s2 ≡ −1 (mod p) has no{5, 8}: the pair {5, 8} yields the two solution. For p − 1 = 4m there has to be the second pair, and this containssolutions of s2 ≡ −1 mod 13. the two solutions of s2 ≡ −1 that we were looking for. Lemma 1 says that every odd prime dividing a number M 2 + 1 must be of the form 4m + 1. This implies that there are inﬁnitely many primes of this form: Otherwise, look at (2 · 3 · 5 · · · qk )2 + 1, where qk is the largest such prime. The same reasoning as above yields a contradiction. Prime ﬁelds If p is a prime, then the set Zp = {0, 1, . . . , p − 1} with addition and multiplication deﬁned “modulo p” forms a ﬁnite ﬁeld. We will need the following simple properties: • For x ∈ Zp , x = 0, the additive inverse (for which we usually + 0 1 2 3 4 write −x) is given by p − x ∈ {1, 2, . . . , p − 1}. If p > 2, then x 0 0 1 2 3 4 and −x are different elements of Zp . 1 1 2 3 4 0 2 2 3 4 0 1 • Each x ∈ Zp {0} has a unique multiplicative inverse x ∈ Zp {0}, 3 3 4 0 1 2 with xx ≡ 1 (mod p). 4 4 0 1 2 3 The deﬁnition of primes implies that the map Zp → Zp , z → xz is injective for x = 0. Thus on the ﬁnite set Zp {0} it must be · 0 1 2 3 4 surjective as well, and hence for each x there is a unique x = 0 0 0 0 0 0 0 with xx ≡ 1 (mod p). 1 0 1 2 3 4 2 0 2 4 1 3 • The squares 02 , 12 , 22 , . . . , h2 deﬁne different elements of Zp , for 3 0 3 1 4 2 h = ⌊ p ⌋. 2 4 0 4 3 2 1 This is since x2 ≡ y 2 , or (x + y)(x − y) ≡ 0, implies that x ≡ yAddition and multiplication in Z5 or that x ≡ −y. The 1 + ⌊ p ⌋ elements 02 , 12 , . . . , h2 are called 2 the squares in Zp . At this point, let us note “on the ﬂy” that for all primes there are solutions for x2 + y 2 ≡ −1 (mod p). In fact, there are ⌊ p ⌋ + 1 distinct squares 2 x2 in Zp , and there are ⌊ p ⌋ + 1 distinct numbers of the form −(1 + y 2 ). 2 These two sets of numbers are too large to be disjoint, since Zp has only p elements, and thus there must exist x and y with x2 ≡ −(1 + y 2 ) (mod p).
- 25. Representing numbers as sums of two squares 19Lemma 2. No number n = 4m + 3 is a sum of two squares. Proof. The square of any even number is (2k)2 = 4k 2 ≡ 0 (mod 4),while squares of odd numbers yield (2k+1)2 = 4(k 2 +k)+1 ≡ 1 (mod 4).Thus any sum of two squares is congruent to 0, 1 or 2 (mod 4).This is enough evidence for us that the primes p = 4m + 3 are “bad.” Thus,we proceed with “good” properties for primes of the form p = 4m + 1. Onthe way to the main theorem, the following is the key step.Proposition. Every prime of the form p = 4m + 1 is a sum of two squares,that is, it can be written as p = x2 + y 2 for some natural numbers x, y ∈ N.We shall present here two proofs of this result — both of them elegant andsurprising. The ﬁrst proof features a striking application of the “pigeon-hole principle” (which we have already used “on the ﬂy” before Lemma 2;see Chapter 25 for more), as well as a clever move to arguments “modulo p”and back. The idea is due to the Norwegian number theorist Axel Thue. √ Proof. Consider the pairs (x′ , y ′ ) of integers with 0 ≤ x′ , y ′ ≤ p, that √ √is, x′ , y ′ ∈ {0, 1, . . . , ⌊ p⌋}. There are (⌊ p⌋ + 1)2 such pairs. Using the √estimate ⌊x⌋ + 1 > x for x = p, we see that we have more than p such √pairs of integers. Thus for any s ∈ Z, it is impossible that all the values For p = 13, ⌊ p⌋ = 3 we considerx′ − sy ′ produced by the pairs (x′ , y ′ ) are distinct modulo p. That is, for x′ , y ′ ∈ {0, 1, 2, 3}. For s = 5, the sumevery s there are two distinct pairs x′ −sy ′ (mod 13) assumes the following √ values: (x′ , y ′ ), (x′′ , y ′′ ) ∈ {0, 1, . . . , ⌊ p⌋}2 ′ Qy 0 1 2 3 x′ Qwith x′ − sy ′ ≡ x′′ − sy ′′ (mod p). Now we take differences: We have 0 0 8 3 11x′ − x′′ ≡ s(y ′ − y ′′ ) (mod p). Thus if we deﬁne x := |x′ − x′′ |, y := 1 1 9 4 12|y ′ − y ′′ |, then we get 2 2 10 5 0 √ 3 3 11 6 1 (x, y) ∈ {0, 1, . . . , ⌊ p⌋}2 with x ≡ ±sy (mod p).Also we know that not both x and y can be zero, because the pairs (x′ , y ′ )and (x′′ , y ′′ ) are distinct.Now let s be a solution of s2 ≡ −1 (mod p), which exists by Lemma 1.Then x2 ≡ s2 y 2 ≡ −y 2 (mod p), and so we have produced (x, y) ∈ Z2 with 0 < x2 + y 2 < 2p and x2 + y 2 ≡ 0 (mod p).But p is the only number between 0 and 2p that is divisible by p. Thusx2 + y 2 = p: done!Our second proof for the proposition — also clearly a Book Proof —was discovered by Roger Heath-Brown in 1971 and appeared in 1984.(A condensed “one-sentence version” was given by Don Zagier.) It is soelementary that we don’t even need to use Lemma 1.Heath-Brown’s argument features three linear involutions: a quite obviousone, a hidden one, and a trivial one that gives “the ﬁnal blow.” The second,unexpected, involution corresponds to some hidden structure on the set ofintegral solutions of the equation 4xy + z 2 = p.
- 26. 20 Representing numbers as sums of two squares Proof. We study the set S := {(x, y, z) ∈ Z3 : 4xy + z 2 = p, x > 0, y > 0}. This set is ﬁnite. Indeed, x ≥ 1 and y ≥ 1 implies y ≤ p and x ≤ p . So 4 4 there are only ﬁnitely many possible values for x and y, and given x and y, there are at most two values for z. 1. The ﬁrst linear involution is given by f : S −→ S, (x, y, z) −→ (y, x, −z), that is, “interchange x and y, and negate z.” This clearly maps S to itself, and it is an involution: Applied twice, it yields the identity. Also, f has no ﬁxed points, since z = 0 would imply p = 4xy, which is impossible. Furthermore, f maps the solutions in T := {(x, y, z) ∈ S : z > 0} to the solutions in ST , which satisfy z < 0. Also, f reverses the signs of T x − y and of z, so it maps the solutions in f U := {(x, y, z) ∈ S : (x − y) + z > 0} to the solutions in SU . For this we have to see that there is no solution with (x−y)+z = 0, but there is none since this would give p = 4xy+z 2 = U 4xy + (x − y)2 = (x + y)2 . What do we get from the study of f ? The main observation is that since f maps the sets T and U to their complements, it also interchanges the elements in T U with these in U T . That is, there is the same number of solutions in U that are not in T as there are solutions in T that are not in U — so T and U have the same cardinality. 2. The second involution that we study is an involution on the set U : g : U −→ U, (x, y, z) −→ (x − y + z, y, 2y − z). First we check that indeed this is a well-deﬁned map: If (x, y, z) ∈ U , then x − y + z > 0, y > 0 and 4(x − y + z)y + (2y − z)2 = 4xy + z 2 , so g(x, y, z) ∈ S. By (x − y + z) − y + (2y − z) = x > 0 we ﬁnd that indeed g(x, y, z) ∈ U . Also g is an involution: g(x, y, z) = (x − y + z, y, 2y − z) is mapped by g to ((x − y + z) − y + (2y − z), y, 2y − (2y − z)) = (x, y, z). g And ﬁnally: g has exactly one ﬁxed point: (x, y, z) = g(x, y, z) = (x − y + z, y, 2y − z) holds exactly if y = z: But then p = 4xy + y 2 = (4x + y)y, which holds only for y = 1 = z, and x = p−1 . 4 But if g is an involution on U that has exactly one ﬁxed point, then the U cardinality of U is odd.
- 27. Representing numbers as sums of two squares 213. The third, trivial, involution that we study is the involution on T thatinterchanges x and y: h : T −→ T, (x, y, z) −→ (y, x, z).This map is clearly well-deﬁned, and an involution. We combine now ourknowledge derived from the other two involutions: The cardinality of T is Tequal to the cardinality of U , which is odd. But if h is an involution ona ﬁnite set of odd cardinality, then it has a ﬁxed point: There is a point On a ﬁnite set of odd cardinality, every(x, y, z) ∈ T with x = y, that is, a solution of involution has at least one ﬁxed point. p = 4x2 + z 2 = (2x)2 + z 2 .Note that this proof yields more — the number of representations of p inthe form p = x2 + (2y)2 is odd for all primes of the form p = 4m + 1. (Therepresentation is actually unique, see [3].) Also note that both proofs arenot effective: Try to ﬁnd x and y for a ten digit prime! Efﬁcient ways to ﬁndsuch representations as sums of two squares are discussed in [1] and [7].The following theorem completely answers the question which started thischapter.Theorem. A natural number n can be represented as a sum of two squaresif and only if every prime factor of the form p = 4m + 3 appears with aneven exponent in the prime decomposition of n. Proof. Call a number n representable if it is a sum of two squares, thatis, if n = x2 + y 2 for some x, y ∈ N0 . The theorem is a consequence ofthe following ﬁve facts.(1) 1 = 12 + 02 and 2 = 12 + 12 are representable. Every prime of the form p = 4m + 1 is representable.(2) The product of any two representable numbers n1 = x2 + y1 and n2 = 1 2 x2 + y2 is representable: n1 n2 = (x1 x2 + y1 y2 ) + (x1 y2 − x2 y1 )2 . 2 2 2(3) If n is representable, n = x2 + y 2 , then also nz 2 is representable, by nz 2 = (xz)2 + (yz)2 .Facts (1), (2) and (3) together yield the “if” part of the theorem.(4) If p = 4m + 3 is a prime that divides a representable number n = x2 + y 2 , then p divides both x and y, and thus p2 divides n. In fact, if we had x ≡ 0 (mod p), then we could ﬁnd x such that xx ≡ 1 (mod p), multiply the equation x2 + y 2 ≡ 0 by x2 , and thus obtain 1 + y 2 x2 = 1 + (xy)2 ≡ 0 (mod p), which is impossible for p = 4m + 3 by Lemma 1.(5) If n is representable, and p = 4m + 3 divides n, then p2 divides n, and n/p2 is representable. This follows from (4), and completes the proof.
- 28. 22 Representing numbers as sums of two squares Two remarks close our discussion: • If a and b are two natural numbers that are relatively prime, then there are inﬁnitely many primes of the form am + b (m ∈ N) — this is a famous (and difﬁcult) theorem of Dirichlet. More precisely, one can show that the number of primes p ≤ x of the form p = am + b is described very 1 x accurately for large x by the function ϕ(a) log x , where ϕ(a) denotes the number of b with 1 ≤ b < a that are relatively prime to a. (This is a substantial reﬁnement of the prime number theorem, which we had discussed on page 10.) • This means that the primes for ﬁxed a and varying b appear essentially at the same rate. Nevertheless, for example for a = 4 one can observe a rather subtle, but still noticeable and persistent tendency towards “more” primes of the form 4m+3: If you look for a large random x, then chances are that there are more primes p ≤ x of the form p = 4m + 3 than of the form p = 4m + 1. This effect is known as “Chebyshev’s bias”; see Riesel [4] and Rubinstein and Sarnak [5]. References [1] F. W. C LARKE , W. N. E VERITT, L. L. L ITTLEJOHN & S. J. R. VORSTER : H. J. S. Smith and the Fermat Two Squares Theorem, Amer. Math. Monthly 106 (1999), 652-665. [2] D. R. H EATH -B ROWN : Fermat’s two squares theorem, Invariant (1984), 2-5. [3] I. N IVEN & H. S. Z UCKERMAN : An Introduction to the Theory of Numbers, Fifth edition, Wiley, New York 1972. [4] H. R IESEL : Prime Numbers and Computer Methods for Factorization, Second edition, Progress in Mathematics 126, Birkhäuser, Boston MA 1994. [5] M. RUBINSTEIN & P. S ARNAK : Chebyshev’s bias, Experimental Mathematics 3 (1994), 173-197. [6] A. T HUE : Et par antydninger til en taltheoretisk metode, Kra. Vidensk. Selsk. Forh. 7 (1902), 57-75. [7] S. WAGON : Editor’s corner: The Euclidean algorithm strikes again, Amer. Math. Monthly 97 (1990), 125-129. [8] D. Z AGIER : A one-sentence proof that every prime p ≡ 1 (mod 4) is a sum of two squares, Amer. Math. Monthly 97 (1990), 144.
- 29. The law of quadratic reciprocity Chapter 5Which famous mathematical theorem has been proved most often? Pythago-ras would certainly be a good candidate or the fundamental theorem of al-gebra, but the champion is without doubt the law of quadratic reciprocityin number theory. In an admirable monograph Franz Lemmermeyer listsas of the year 2000 no fewer than 196 proofs. Many of them are of courseonly slight variations of others, but the array of different ideas is still im-pressive, as is the list of contributors. Carl Friedrich Gauss gave the ﬁrstcomplete proof in 1801 and followed up with seven more. A little laterFerdinand Gotthold Eisenstein added ﬁve more — and the ongoing list ofprovers reads like a Who is Who of mathematics.With so many proofs present the question which of them belongs in theBook can have no easy answer. Is it the shortest, the most unexpected, orshould one look for the proof that had the greatest potential for general-izations to other and deeper reciprocity laws? We have chosen two proofs(based on Gauss’ third and sixth proofs), of which the ﬁrst may be the sim-plest and most pleasing, while the other is the starting point for fundamental Carl Friedrich Gaussresults in more general structures.As in the previous chapter we work “modulo p”, where p is an odd prime;Zp is the ﬁeld of residues upon division by p, and we usually (but not al-ways) take these residues as 0, 1, . . . , p − 1. Consider some a ≡ 0 (mod p),that is, p ∤ a. We call a a quadratic residue modulo p if a ≡ b2 (mod p) forsome b, and a quadratic nonresidue otherwise. The quadratic residues aretherefore 12 , 22 , . . . , ( p−1 )2 , and so there are p−1 quadratic residues and 2 2 For p = 13, the quadratic residues arep−1 p−1 2 quadratic nonresidues. Indeed, if i ≡ j (mod p) with 1 ≤ i, j ≤ 2 , 2 2 12 ≡ 1, 22 ≡ 4, 32 ≡ 9, 42 ≡ 3,then p | i2 − j 2 = (i − j)(i + j). As 2 ≤ i + j ≤ p − 1 we have p | i − j, 52 ≡ 12, and 62 ≡ 10; the nonresiduesthat is, i ≡ j (mod p). are 2, 5, 6, 7, 8, 11.At this point it is convenient to introduce the so-called Legendre symbol.Let a ≡ 0 (mod p), then a 1 if a is a quadratic residue, := p −1 if a is a quadratic nonresidue.The story begins with Fermat’s “little theorem”: For a ≡ 0 (mod p), ap−1 ≡ 1 (mod p). (1)In fact, since Z∗ = Zp {0} is a group with multiplication, the set {1a, 2a, p3a, . . . , (p − 1)a} runs again through all nonzero residues, (1a)(2a) · · · ((p − 1)a) ≡ 1 · 2 · · · (p − 1) (mod p),and hence by dividing by (p − 1)!, we get ap−1 ≡ 1 (mod p).
- 30. 24 The law of quadratic reciprocity In other words, the polynomial xp−1 − 1 ∈ Zp [x] has as roots all nonzero residues. Next we note that p−1 p−1 xp−1 − 1 = (x 2 − 1)(x 2 + 1). Suppose a ≡ b2 (mod p) is a quadratic residue. Then by Fermat’s little p−1 theorem a 2 ≡ bp−1 ≡ 1 (mod p). Hence the quadratic residues are p−1 precisely the roots of the ﬁrst factor x 2 − 1, and the p−1 nonresidues 2 p−1 must therefore be the roots of the second factor x 2 + 1. Comparing this to the deﬁnition of the Legendre symbol, we obtain the following important tool.For example, for p = 17 and a = 3 wehave 38 = (34 )2 = 812 ≡ (−4)2 ≡ Euler’s criterion. For a ≡ 0 (mod p),−1 (mod 17), while for a = 2 we get a p−128 = (24 )2 ≡ (−1)2 ≡ 1 (mod 17). ≡ a 2 (mod p).Hence 2 is a quadratic residue, while 3 pis a nonresidue. This gives us at once the important product rule ab a b = , (2) p p p since this obviously holds for the right-hand side of Euler’s criterion. The product rule is extremely helpful when one tries to compute Legendre sym- bols: Since any integer is a product of ±1 and primes we only have to q compute ( −1 ), ( p ), and ( p ) for odd primes q. p 2 By Euler’s criterion ( −1 ) = 1 if p ≡ 1 (mod 4), and ( −1 ) = −1 if p ≡ p p 3 (mod 4), something we have already seen in the previous chapter. The 2 2 case ( p ) will follow from the Lemma of Gauss below: ( p ) = 1 if p ≡ 2 ±1 (mod 8), while ( p ) = −1 if p ≡ ±3 (mod 8). Gauss did lots of calculations with quadratic residues and, in particular, studied whether there might be a relation between q being a quadratic residue modulo p and p being a quadratic residue modulo q, when p and q are odd primes. After a lot of experimentation he conjectured and then proved the following theorem. Law of quadratic reciprocity. Let p and q be different odd primes. Then q p p−1 q−1 ( )( ) = (−1) 2 2 . p q If p ≡ 1 (mod 4) or q ≡ 1 (mod 4), then p−1 (resp. q−1 ) is even, and 2 2 p−1 q−1 therefore (−1) 2 2 = 1; thus ( p ) = ( p ). When p ≡ q ≡ 3 (mod 4), we q qExample: ( 17 ) = ( 17 ) = ( 2 ) = −1, 3 3 3 have ( p ) = −( p ). Thus for odd primes we get ( p ) = ( q ) unless both p q q q pso 3 is a nonresidue mod 17. and q are congruent to 3 (mod 4).
- 31. The law of quadratic reciprocity 25First proof. The key to our ﬁrst proof (which is Gauss’ third) is a countingformula that soon came to be called the Lemma of Gauss: Lemma of Gauss. Suppose a ≡ 0 (mod p). Take the numbers 1a, 2a, . . . , p−1 a and reduce them modulo p to the residue system 2 smallest in absolute value, ia ≡ ri (mod p) with − p−1 ≤ ri ≤ p−1 2 2 for all i. Then a ( ) = (−1)s , where s = #{i : ri < 0}. p Proof. Suppose u1 , . . . , us are the residues smaller than 0, and thatv1 , . . . , v p−1 −s are those greater than 0. Then the numbers −u1 , . . . , −us 2are between 1 and p−1 , and are all different from the vj s (see the margin); 2 If −ui = vj , then ui + vj ≡ 0 (mod p).hence {−u1 , . . . , −us , v1 , . . . , v p−1 −s } = {1, 2, . . . , p−1 }. Therefore 2 Now ui ≡ ka, vj ≡ ℓa (mod p) implies 2 p | (k + ℓ)a. As p and a are relatively p−1 prime, p must divide k + ℓ which is im- (−ui ) vj = 2 !, i j possible, since k + ℓ ≤ p − 1.which implies p−1 (−1)s ui vj ≡ 2 ! (mod p). i jNow remember how we obtained the numbers ui and vj ; they are theresidues of 1a, · · · , p−1 a. Hence 2 p−1 p−1 p−1 2 ! ≡ (−1)s ui vj ≡ (−1)s 2 !a 2 (mod p). i j p−1Cancelling 2 ! together with Euler’s criterion gives a p−1 ( ) ≡ a 2 ≡ (−1)s (mod p), pand therefore ( a ) = (−1)s , since p is odd. pWith this we can easily compute ( p ): Since 1 · 2, 2 · 2, . . . , p−1 · 2 are all 2 2between 1 and p − 1, we have p−1 p−1 p−1 s = #{i : 2 < 2i ≤ p − 1} = 2 − #{i : 2i ≤ 2 } = ⌈ p−1 ⌉. 4Check that s is even precisely for p = 8k ± 1.The Lemma of Gauss is the basis for many of the published proofs of thequadratic reciprocity law. The most elegant may be the one suggestedby Ferdinand Gotthold Eisenstein, who had learned number theory fromGauss’ famous Disquisitiones Arithmeticae and made important contribu-tions to “higher reciprocity theorems” before his premature death at age 29.His proof is just counting lattice points!

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