Basic Number Theory. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. Last revision: June 11, 2001

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1 Basc Number Theory Instructor: Laszlo Baba Notes by Vncent Lucarell and the nstructor Last revson: June, 200 Notaton: Unless otherwse stated, all varables n ths note are ntegers. For n 0, [n] = {, 2,..., n}. The formula d n denotes the relaton d dvdes n,. e., ( k(n = dk. We also say d s a dvsor of n or n s a multple of d. Note that ( a(a a, ncludng 0 0 (even though we do not allow dvson by zero!. In fact 0 n n = 0. Note also that ( n(( k(n k k = ±. We wrte a b (mod m f m a b ( a s congruent to b modulo m. Gcd, congruences Exercse. Prove that the product of n consecutve ntegers s always dvsble by n!. Hnt. One-lne proof. Exercse.2 (The Dvsor Game Select an nteger n 2. Two players alternate namng postve dvsors of n subject to the followng rule: no dvsor of any prevously named nteger can be named. The frst player forced to name n loses. Example: f n = 30 then the followng s a possble sequence of moves: 0, 3, 6, 5, at whch pont t s the frst player s move; he s forced to say 30 and loses.. Fnd a wnnng strategy for the frst player when n s a prme power or the product of two prme powers or when n s square-free (n s not dvsble by the square of any prme. 2. Prove: n 2, the frst player has a wnnng strategy. (Hnt: prove, n two or three lnes, the exstence of a wnnng strategy. Let Dv (n denote the set of postve dvsors or n. Exercse.3 Prove: ( a, b( d(dv (a Dv (b = Dv (d. A nonnegatve d satsfyng ths statement s called the g.c.d. of a and b. Note that g.c.d. (a, b = 0 a = b = 0. Defne l.c.m. analogously. When s l.c.m. (a, b = 0?

2 Exercse.4 Prove: g.c.d. (a k, a l = a d, where d = g.c.d. (k, l. Defnton.5 The Fbonacc numbers are defned by the recurrence F n = F n + F n 2, F 0 = 0, F =. Exercse.6 Prove: g.c.d. (F k, F l = F d, where d = g.c.d. (k, l. Exercse.7 Prove: f a b (mod m then g.c.d. (a, m = g.c.d. (b, m. Exercse.8 Prove: f a, b 0 then g.c.d. (a, b l.c.m. (a, b = ab. Exercse.9 Prove: congruence modulo m s an equvalence relaton on Z. The equvalence classes are called the resdue classes mod m. There are m resdue classes modulo m. Under the natural operatons they form the rng Z/mZ. The addtve group of ths rng s cyclc. Exercse.0 Prove that the sequence of Fbonacc numbers modm s perodc. The length of the perod s m 2. Exercse. An nteger-preservng polynomal s a polynomal f(x such that ( a Z(f(a Z. Prove that f(x s nteger-preservng f and only f t can be wrtten as f(x = n ( x a =0 wth sutable nteger coeffcents a. Here ( x x(x... (x + = ;! ( x =. 0 Exercse.2 A congruence-preservng polynomal s an nteger-preservng polynomal such that ( a, b, m Z(a b (mod m f(a f(b (mod m. Prove that f(x s congruencepreservng f and only f ( (e a n the expresson (, where e = l.c.m. (, 2,...,. Exercse.3 A multplcatve nverse of a modulo m s an nteger x such that ax (mod m; notaton: x = a mod m. Prove: a mod m g.c.d. (a, m =. Exercse.4 (Wlson s theorem Prove: (p! (mod p. Hnt: match each number wth ts multplcatve nverse n the product (p! Exercse.5 Prove: f g.c.d. (a, p = then p j= j p (a. Hnt. Match terms on the rght hand sde wth terms on the left hand sde so that correspondng terms satsfy j a (mod p. Exercse.6 Infer Fermat s lttle Theorem from the precedng exercse: f g.c.d. (a, p = then a p (mod p. ( 2

3 Exercse.7 Use the same dea to prove the Euler Fermat theorem: f g.c.d. (a, m = then a ϕ(m (mod m. (ϕ s Euler s ϕ functon, see below. Exercse.8 Prove: f p s a prme and f s a polynomal wth nteger coeffcents then f(x p f(x p (mod p. Here the congruence of two polynomals means coeffcentwse congruence. The multplcatve group (Z/mZ conssts of the mod m resdue classes relatvely prme to m. Its order s ϕ(m. Exercse +.9 Prove: f p s a prme then (Z/pZ s cyclc. A generator of ths group s called a prmtve root mod p. Exercse +.20 Prove: f p s an odd prme then (Z/p k Z s cyclc. Exercse +.2 If k 2 then the group (Z/2 k Z s not cyclc but the drect sum of a cyclc group of order 2 and a cyclc group of order 2 k 2. 2 Arthmetc Functons Defnton 2. (Euler s Ph Functon ϕ(n = { k [n] : g.c.d. (k, n = } = number of postve ntegers not greater than n whch are relatvely prme to n Exercse 2.2 Show that the number of complex prmtve n-th roots of unty s ϕ(n. Show that f d n then the number of elements of order d n a cyclc group of order n s ϕ(d. Exercse 2.3 Show ϕ(d = n. d n Exercse Let D n = (d j denote the n n matrx wth d j = g.c.d. (, j. Prove: det D n = ϕ(ϕ(2 ϕ(n. (Hnt. Let Z = (z j be the matrx wth z j = f j and z j = 0 otherwse. Consder the matrx Z T F Z where F s the dagonal matrx wth entres ϕ(,..., ϕ(n and Z T s Ztranspose (reflecton n the man dagonal. Defnton 2.5 (Number of [postve] dvsors d(n = {d N : d n} 3

4 Exercse 2.6 Prove: d(n < 2 n. Exercse Prove: ( ɛ > 0( n 0 ( n > n 0 (d(n < n ɛ. (Hnt. Use a consequence of the Prme Number Theorem (see the next secton. Prove that d(n < n c/ ln ln n for some constant c. The best asymptotc constant s c = ln 2 + o(. Exercse Prove that for nfntely many values of n the reverse nequalty d(n > nc/ ln ln n holds (wth another constant c > 0. (Agan, use the PNT. Exercse Let D(n = (/n n d( (the average number of dvsors. Prove: D(n ln(n. (Comment. If we pck an nteger t at random between and n then D(n wll be the expected number of dvsors of t. Make your proof very smple (3 lnes. Do not use the PNT. Exercse Prove: (/n n d(2 = Θ((ln n 3. Defnton 2. (Sum of [postve] dvsors σ(n = d n d Defnton 2.2 Let n = p k pkr r where the p are dstnct prmes and k > 0. Set ν(n = r (number of dstnct prme dvsors; so ν( = 0. Set ν (n = k + + k r (total number of prme dvsors; so ν ( = 0. Exercse Prove that the expected number of dstnct prme dvsors of a random nteger [n] s asymptotcally ln ln n : n ν( ln ln n. n How much larger s ν? On average, not much. Prove that the average value of ν s also asymptotc to ln ln n. Termnology. n s square-free f ( p prme (p 2 n. Defnton 2.4 (Möbus Functon n = µ(n = ( k n = p p k where the p are dstnct (n s square-free 0 f ( p(p 2 n Exercse 2.5 Let δ(n = d n µ(d. Evaluate δ(n. Defnton 2.6 For s > defne the zeta functon ζ(s = 4 n= n s.

5 Exercse 2.7 Prove Euler s dentty: ζ(s = p prme p s. Exercse 2.8 Prove: Exercse 2.9 Prove: Exercse 2.20 Prove: ζ(s = (ζ(s 2 = n= n= ζ(s(ζ(s = µ(n n s. d(n n s. n= σ(n n s. Exercse 2.2 Prove: ζ(2 = π 2 /6. Exercse 2.22 Gve a natural defnton whch wll make followng statement sensble and true: the probablty that a random postve nteger n satsfes n 3 (mod 7 s /7. Our choce of a random postve nteger should be unform (obvously mpossble. (Hnt. Consder the ntegers up to x; then take the lmt as x. Exercse 2.23 Make sense out of the queston What s the probablty that two random postve ntegers are relatvely prme? Prove that the answer s 6/π 2. Hnt. To prove that the requred lmt exsts may be somewhat tedous. If you want to see the fun part, assume the exstence of the lmt, and prove n just two lnes that the lmt must be /ζ(2. Defnton 2.24 Let F be a feld. f : N F s called multplcatve f f s called completely multplcatve f f s called addtve f Exercse 2.25 Show that ( a, b(g.c.d. (a, b = f(ab = f(af(b. ( a, b(f(ab = f(af(b. ( a, b(g.c.d. (a, b = f(ab = f(a + f(b.. ϕ, σ, d, and µ are multplcatve but not completely multplcatve 5

6 2. ν s addtve and ν s completely addtve. Exercse 2.26 Show. ϕ ( p k = p k p k = (p p k 2. d ( p k = k + 3. σ ( p k = pk+ p Exercse 2.27 Show ( r r. ϕ = (p p k 2. d 3. σ ( r ( r p k p k p k = = r (k + r p k + p Exercse 2.28 Show ϕ(n = n p N p prme ( p Let F be a feld and f : N F. Defne g(n = d n f(d. Exercse 2.29 (Möbus Inverson Formula Show f(n = ( n g(dµ. d d N Exercse 2.30 Use the Möbus Inverson Formula together wth Exercse 2.3 for a second proof of Exercse Exercse 2.3 Prove that the sum of the complex prmtve n-th roots of unty s µ(n. Defnton 2.32 The n-th cyclotomc polynomal Φ n (x s defned as Φ n (x = ω (x ω where the product ranges over all complex prmtve n-th roots of unty. Note that the degree of Φ n (x s ϕ(n. Also note that Φ (x = x, Φ 2 (x = x+, Φ 3 (x = x 2 +x+, Φ 4 (x = x 2 +, Φ 5 (x = x 4 + x 3 + x 2 + x +, Φ 6 (x = x 2 x +. 6

7 Exercse 2.33 Prove that Φ n (x has nteger coeffcents. What s the coeffcent of x ϕ(n? Exercse 2.34 Prove: f p s a prme then Φ p (x = x p + x p x +. Exercse 2.35 Prove: Φ n (x = d n(x d µ(n/d. Exercse (Bateman Let A n denote the sum of the absolute values of the coeffcents of Φ n (x. Prove that A n < n d(n/2. Infer from ths that A n < exp(n c/ ln ln n for some constant c. Hnt: We say that the power seres n=0 a nx n domnates the power seres n=0 b nx n f ( n( b n a n. Prove that the power seres domnates Φ n (x. d n x d Note: Erdős proved that ths bound s tght, apart from the value of the constant: for nfntely many values of n, A n > exp(n c/ ln ln n for another constant c > 0. Exercse (Hermte Let f(x = n =0 a x be a monc polynomal of degree n (. e., a n = wth nteger coeffcents. Suppose all roots of f have unt absolute value. Prove that all roots of f are roots of unty. (In other words, f all algebrac conjugates of a complex algebrac number z have unt absolute value then z s a root of unty. 3 Prme Numbers Exercse 3. Prove: n = ln n + O(. Exercse 3.2 Prove: p x /p =, where the product s over all prmes x and the summaton extends over all postve ntegers wth no prme dvsors greater than x. In partcular, the sum on the rght-hand sde converges. It also follows that the left-hand sde s greater than ln x. Exercse 3.3 Prove: /p =. (Hnt. Use the precedng exercse. Take natural logarthms; use the power seres expanson of ln( z. Conclude that p x /p > ln ln x + O(. (In other words, p x /p ln ln x s bounded from below. 7

8 Exercse Prove: bounded. p x /p = ln ln x + O(. (In other words, p x /p ln ln x s ( n Exercse Prove ϕ(n = Ω and fnd the largest mplct asymptotc constant. ln ln n Let π(x the number of prmes less than or equal to x. Theorem 3.6 (Prme Number Theorem, Hadamard and de la Vallée Poussn, 896 π(x x ln x p n+ Exercse 3.7 Use the PNT to show that lm n p n =, where p n s the n-th prme. Exercse 3.8 Use the PNT to prove p n n ln n. Exercse 3.9 Prove p = exp ( x( + o(. Prove that ths result s n fact equvalent to the PNT. p x p prme Exercse 3.0 Let e n = l.c.m. (, 2,..., n. Prove: e n = exp ( n( + o(. Prove that ths result s n fact equvalent to the PNT. Exercse 3. Prove: p x p x2 /(2 ln x. (Use the PNT. Defnton 3.2 A permutaton s a bjecton of a set to tself. The permutatons of a set form a group under composton. The symmetrc group of degree n s the group of all permutatons of a set of n elements; t has order n!. The exponent of a group s the l.c.m. of the orders of all elements of the group. Exercse 3.3 Prove: the exponent of S n s e n. Exercse Let m(n denote the maxmum of the orders of the elements n S n. Prove: m(n = exp( n ln n( + o(. Exercse 3.5 Let a(n denote the typcal order of elements n S n. Prove that ln a(n = O((ln n 2. ( Typcal order means that 99% of the elements has order fallng n the stated range. Here 99 s arbtrarly close to 00. Hnt. Prove that a typcal permutaton has O(ln n cycles. Erdős and Turán proved n 965 that n fact ln a(n (ln n 2 /2. Exercse 3.6 Prove from frst prncples: p<x p prme 8 p < 4 x. (Hnt: f n < p 2n then p ( 2n n.

9 Exercse 3.7 Prove: f p > 2n then p 2 ( 2n n. Exercse 3.8 Prove: f q s a prme power dvdng ( 2n n then q n. (Hnt. Gve a formula for the hghest exponent of a prme p whch dvdes ( 2n n. Frst, fnd a formula for the exponent of p n n!. Exercse 3.9 Prove from frst prncples: p<x p prme p > (2 + o( x. (Hnt. Consder the prmepower decomposton of ( x x/2. Show that the contrbuton of the powers of prmes x s neglgble. Exercse 3.20 Paul Erdős was an undergraduate when he found a smple proof of Chebyshev s theorem based on the prme factors of ( 2n n. Chebyshev s theorem s a precursor of the PNT; t says that ( x π(x = Θ. ln x Followng Erdős, prove Chebyshev s Theorem from frst prncples. The proof should be only a few lnes, based Exercses 3.6 and Quadratc Resdues Defnton 4. a s a quadratc resdue modp f (p a and ( b(a b 2 mod p. Exercse 4.2 Prove: a s a quadratc resdue modp a (p /2 (mod p. Defnton 4.3 a s a quadratc non-resdue modp f ( b(a b 2 mod p. Exercse 4.4 Prove: a s a quadratc non-resdue modp a (p /2 (mod p. Defnton 4.5 (Legendre Symbol ( a f a s a quadratc resdue modp = f a s a quadratc non-resdue modp p 0 f p a Let F q be a fnte feld of odd prme power order q. Defnton 4.6 a F q s a quadratc resdue f a 0 and ( b(a = b 2. Exercse 4.7 Prove: a s a quadratc resdue n F q a (q /2 =. Defnton 4.8 a F q s a quadratc non-resdue f ( b(a b 2. 9

10 Exercse 4.9 Prove: a s a quadratc non-resdue n F q a (q /2 =. Exercse 4.0 Prove: n F q, the number of quadratc resdues equals the number of quadratc non-resdues; so there are (q /2 of each. (As before, q s an odd prme power. Defnton 4. Let q be an odd prme power. We defne the quadratc character χ: F q {0,, } C by f a s a quadratc resdue χ(a = f a s a non-resdue 0 f a = 0 Note that f q = p (.e. prme and not prme power then χ(a = Exercse 4.2 Prove χ s multplcatve. ( a. p Exercse 4.3 The Legendre Symbol s completely multplcatve n the numerator. Exercse 4.4 Prove that s a quadratc resdue n F q f and only f q (mod 4. Exercse 4.5 Prove that a F q χ(a(a =. Hnt. Dvde by a 2. Exercse 4.6 Prove that each of the four pars (±, ± occur a roughly equal number of tmes ( q/4 as (χ(a, χ(a (a F q. Roughly equal means the dfference s bounded by a small constant. Moral: for a random element a F q, the values of χ(a and χ(a are nearly ndependent. Exercse 4.7 Let f(x = ax 2 +bx+c be a quadratc polynomal over F q (a, b, c F q, a 0. Prove: f b 2 4ac 0 then a F q χ(f(a 2. What happens f b 2 4ac = 0? Exercse 4.8 a 2 + b 2 (mod 4. Exercse 4.9 (Gauss Prove: a prme p can be wrtten as the sum of two squares f and only f p = 2 or p (mod 4. Hnt. The ( necessty s clear from the precedng exercse. For suffcency, assume p (mod 4. Then = and therefore ( a(p a 2 +. Consder the lattce (plane grd L Z 2 p consstng of all ntegral lnear combnatons of the vectors (a, and (p, 0. Observe that f (x, y L then p x 2 + y 2. Moreover, the area of the fundamental parallelogram of the lattce s p. Apply Mnkowsk s Theorem (below to ths lattce to obtan a nonzero lattce pont (x, y satsfyng x 2 + y 2 < 2p. (Ths proof s due to P. Turán. 0

11 5 Lattces and dophantne approxmaton Defnton 5. An n-dmensonal lattce (grd s the set L of all ntegral lnear combnatons n a b of a bass {b,..., b n } of R n (a Z. The set of those lnear combnatons wth 0 a (a R form a fundamental parallelopped. Exercse 5.2 The volume of the fundamental parallelopped of the lattce L s det(l := det(b,..., b n. Exercse 5.3 (Mnkowsk s Theorem Let L be an n-dmensonal lattce and let V be the volume of ts fundamental parallelopped. Let A R n be an n-dmensonal convex set, symmetrcal about the orgn (. e., A = A, wth volume greater than 2 n V. Then A L {0},. e., A contans a lattce pont other than the orgn. Hnt. Lnear transformatons don t change the proporton of volumes, and preserve convexty and central symmetry. So WLOG L = Z n wth {b } the standard bass. The fundamental parallelopped s now the unt cube C. Consder the lattce 2L = (2Z n. Then the quotent space R n /(2Z n can be dentfed wth the cube 2C whch has volume 2 n. Snce A has volume > 2 n, there exst two ponts u, v A whch are mapped to the same pont n 2C,. e., all coordnates of u v are even ntegers. Show that (u v/2 A L. Exercse 5.4 Fndng short vectors n a lattce s of partcular mportance. followng corollary to Mnkowsk s Theorem: Prove the ( v L(0 < v (det L /n. Defnton 5.5 Let α,..., α n R. A smultaneous ɛ-approxmaton of the α s a sequence of fractons p /q wth a common denomnator q > 0 such that ( ( qα p ɛ. Exercse (Drchlet ( α,..., α n R( ɛ > 0( an ɛ-approxmaton wth the denomnator satsfyng 0 < q ɛ n. Hnt. Apply the precedng exercse to the (n + -dmensonal lattce L wth bass e,..., e n, f where f = n α e + ɛ n+ e n+ and {e,..., e n+ } s the standard bass.