Abstract. October 12, 2012 Delhi University, South Campus Department of Mathematics. Michel Waldschmidt

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1 Abstract October 2, 202 Delhi University, South Campus Department of Mathematics Problems in number theory are sometimes easy to state and often very hard to solve. We survey some of them. Number theory: Challenges of the twenty first century Michel Waldschmidt Institut de Mathématiques de Jussieu Paris VI miw/ /95 2/95 Extended abstract We start with prime numbers. The twin prime conjecture and the Goldbach conjecture are among the main challenges. Are there infinitely many Mersenne (resp. Fermat) prime numbers? The largest known prime numbers are Mersenne numbers. Mersenne prime numbers are also related with perfect numbers, a problem considered by Euclid and still unsolved. One the main challenges for specialists of number theory is Riemann s hypothesis, which is now more than 50 years old. Diophantine equations conceal plenty of mysteries. Fermat s Last Theorem has been proved by A. Wiles, but many more questions are waiting for an answer. We discuss a conjecture due to S.S. Pillai, aswell as the abc-conjecture of Oesterlé Masser. Kontsevich and Zagier introduced the notion of periods and suggested a far reaching statement which would solve a large number of open problems of irrationality and transcendence. Finally we discuss open problems (initiated by E. Borel in 905 and then in 950) on the decimal (or binary) development of algebraic numbers. Almost nothing is known on this topic. Hilbert s 8th Problem August 8, 900 David Hilbert ( ) Second International Congress of Mathematicians in Paris. Twin primes, Goldbach s Conjecture, Riemann Hypothesis 3/95 4/95

2 The seven Millennium Problems The Clay Mathematics Institute (CMI) Cambridge, Massachusetts 7 million US$ prize fund for the solution to these problems, with million US$ allocated to each of them. Paris, May 24, 2000 : Timothy Gowers, John Tate and Michael Atiyah. Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations PvsNP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory Numbers Numbers =realorcomplexnumbersr, C. Natural integers : N = {0,, 2,...}. Rational integers : Z = {0, ±, ±2,...}. 5/95 6/95 Prime numbers Numbers with exactly two divisors : 2, 3, 5, 7,, 3, 7, 9, 23, 29, 3, 37, 4, 43, 47, 53,... The On-Line Encyclopedia of Integer Sequences Composite numbers Numbers with more than two divisors : 4, 6, 8, 9, 0, 2, 4, 5, 6, 8, 20, 2, 22, 24, 25, 26, 27,... The composite numbers : numbers n of the form x y for x> and y>. 7/95 8/95

3 Euclid of Alexandria (about 325 BC about 265 BC) Twin primes Conjecture : there are infinitely many primes p such that p +2 is prime. Examples : 3, 5, 5, 7,, 3, 7, 9,... More generally : is every even integer (infinitely often) the di erence of two primes? of two consecutive primes? Largest known example of twin primes (October 202) with decimal digits : Given any finite collection p,...,p n of primes, there is one prime which is not in this collection ± 9/95 0 / 95 Goldbach s Conjecture Sums of two primes Leonhard Euler ( ) Letter of Christian Goldbach ( ) to Euler, 742 : any integer 5 is sum of at most 3 primes. Euler : Equivalent to : any even integer greater than 2 can be expressed as the sum of two primes. Proof : 2n 2=p + p 0 ()2n = p + p 0 +2()2n +=p + p =2+2 6=3+3 8=5+3 0=7+3 2 = = = = = = = = / 95 2 / 95

4 Sums of primes Circle method 27 is neither prime nor a sum of two primes Weak (or ternary) Goldbach Conjecture : every odd integer 7 is the sum of three odd primes. Terence Tao, February 4, 202, arxiv: : Every odd number greater than is the sum of at most five primes. H. A. Helfgott, May 23, 202, arxiv: v Minor arcs for Goldbach s problem. Srinivasa Ramanujan ( ) G.H. Hardy ( ) J.E. Littlewood ( ) Hardy, ICM Stockholm, 96 Hardy and Ramanujan (98) : partitions Hardy and Littlewood ( ) : Some problems in Partitio Numerorum 3 / 95 4 / 95 Circle method Largest explicitly known prime numbers Hardy and Littlewood Ivan Matveevich Vinogradov (89 983) August 23, decimal digits June 3, decimal digits Every su ciently large odd integer is the sum of at most three primes. September 6, decimal digits / 95 6 / 95

5 Large prime numbers Marin Mersenne The nine largest explicitly known prime numbers are of the form 2 p. One knows (as of October 2, 202) 54 prime numbers with more than decimal digits 262 prime numbers with more than decimal digits List of the 5000 largest explicitly known prime numbers : 47 prime numbers of the form of the form 2 p are known / 95 8 / 95 Mersenne prime numbers Mersenne prime numbers If a number of the form 2 k is prime, then k itself is prime. In 536, Hudalricus Regius noticed that 2 aprimenumber:2047= =2047is not A prime number of the form 2 p is called a Mersenne prime. 47 of them are known, among them the 9 largest are also the largest explicitly known primes. The smallest Mersenne primes are 3=2 2, 7=2 3 3 = 2 5, 27 = 2 7. Are there infinitely many Mersenne primes? In the preface of Cogitata Physica-Mathematica (644), Mersenne claimed that the numbers 2 n are prime for n =2, 3, 5, 7, 3, 7, 9, 3, 67, 27 and 257 and that they are composite for all other values of n<257. The correct list is 2, 3, 5, 7, 3, 7, 9, 3, 6, 89, 07 and 27. http ://oeis.org/a / / 95

6 Perfect numbers A number if perfect if it is equal to the sum of it divisors, excluding itself. For instance 6 is the sum +2+3, and the divisors of 6 are, 2, 3 and 6. In the same way, the divisors of 28 are, 2, 4, 7, 4 and 28. The sum is 28, hence 28 is perfect. Notice that 6=2 3 and 3 is a Mersenne prime 2 2. Also 28 = 4 7 and 7 is a Mersenne prime 2 3. Other perfect numbers : 496 = 6 3 with 6 = 2 4, 3 = 2 5, 828 = and 64 = 2 6, 27 = 2 7,... Perfect numbers Euclid, Elements, Book IX : numbers of the form 2 p (2 p ) with 2 p a(mersenne) prime(hencep is prime) are perfect. Euler : all even perfect numbers are of this form. Sequence of perfect numbers : 6, 28, 496, 828, ,... Are there infinitely many even perfect number? Does there exist odd perfect numbers? 2 / / 95 Fermat numbers Fermat numbers are the numbers F n =2 2n +. Fermat primes F 0 =3, F =5, F 2 =7, F 3 =257, F 4 =65537are prime They are related with the construction of regular polygons with ruler and compass. Fermat suggested in 650 that all F n are prime Euler : F 5 =2 32 +is divisible by = Pierre de Fermat (60 665) 64 = = Are there infinitely many Fermat primes? Only five are known. 23 / / 95

7 Leonhard Euler ( ) Johann Carl Friedrich Gauss ( ) Let p n be the n-th prime. Gauss introduces For s>, (s) = X n = Y s n p For s =: X p p =+. ( p s ). 25 / 95 Problem : estimate from above E(x) = (x) (x) = X papplex He observes numerically (t + dt) (t) dt log t Define the density d by Z x 0 (x) = Z x 0 d (t). dt log t. 26 / 95 Plot Lejeune Dirichlet ( ) 837 : For gcd(a, q) =, X p a (mod q) p =+. 27 / / 95

8 Pafnuty Lvovich Chebyshev (82 894) Riemann : 0.8 x log x apple (x) apple.2 x log x (s) =0 with 0 < <e(s) < implies <e(s) =/2. 29 / / 95 Riemann Hypothesis Certainly one would wish for a stricter proof here ; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. (Monatsberichte der Berliner Akademie, November 859) Bernhard Riemann s Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, herausgegeben under Mitwirkung von Richard Dedekind, von Heinrich Weber. (Leipzig : B. G. Teubner 892) Small Zeros Zeta Infinitely many zeroes on the critical line : Hardy 94 First 0 3 zeroes : Gourdon Demichel 3 / / 95

9 Riemann Hypothesis Riemann Hypothesis is equivalent to : E(x) apple Cx /2 log x Prime Number Theorem : (x) ' x/ log x Jacques Hadamard Charles de la Vallée Poussin ( ) ( ) for the remainder E(x) = (x) Z x 0 dt log t. Let Even(N) (resp. Odd(N)) denotethenumberofpositive integers apple N with an even (resp. odd) number of prime factors, counting multiplicities. Riemann Hypothesis is also equivalent to Even(N) Odd(N) apple CN / : ( + it) 6= 0for t 2 R \{0}. 33 / / 95 Diophantine Problems Diophantus of Alexandria (250 ±50) Fermat s Last Theorem x n + y n = z n Pierre de Fermat Andrew Wiles Solution in / / 95

10 S.Sivasankaranarayana Pillai (90 950) Collected works of S. S. Pillai, ed. R. Balasubramanian and R. Thangadurai, 200. Square, cubes... A perfect power is an integer of the form a b where a and b> are positive integers. Squares :, 4, 9, 6, 25, 36, 49, 64, 8, 00, 2, 44, 69, 96,... Cubes :, 8, 27, 64, 25, 26, 343, 52, 729, 000, 33,... http :// kr/pillai.html Fifth powers :, 32, 243, 024, 325, 7776, 6 807, , / / 95 Perfect powers, 4, 8, 9, 6, 25, 27, 32, 36, 49, 64, 8, 00, 2, 25, 28, 44, 69, 96, 26, 225, 243, 256, 289, 324, 343, 36, 400, 44, 484, 52, 529, 576, 625, 676, 729, 784,... Consecutive elements in the sequence of perfect powers Di erence : (8, 9) Di erence 2 : (25, 27),... Di erence 3 : (, 4), (25, 28),... Di erence 4 : (4, 8), (32, 36), (2, 25),... Neil J. A. Sloane s encyclopaedia http ://oeis.org/a / 95 Di erence 5 : (4, 9), (27, 32), / 95

11 Two conjectures Pillai s Conjecture : Subbayya Sivasankaranarayana Pillai (90-950) Eugène Charles Catalan (84 894) Catalan s Conjecture : In the sequence of perfect powers, 8, 9 is the only example of consecutive integers. Pillai s Conjecture : In the sequence of perfect powers, the di erence between two consecutive terms tends to infinity. Pillai s Conjecture : In the sequence of perfect powers, the di erence between two consecutive terms tends to infinity. Alternatively : Let k be a positive integer. The equation x p y q = k, where the unknowns x, y, p and q take integer values, all 2, has only finitely many solutions (x, y, p, q). 4 / / 95 Pillai s conjecture Pillai, S. S. On the equation 2 x 3 y =2 X +3 Y, Bull. Calcutta Math. Soc. 37, (945) ItakethisopportunitytoputinprintaconjecturewhichI gave during the conference of the Indian Mathematical Society held at Aligarh. Arrange all the powers of integers like squares, cubes etc. in increasing order as follows :, 4, 8, 9, 6, 25, 27, 32, 36, 49, 64, 8, 00, 2, 25, 28,... Let a n be the n-th member of this series so that a =, a 2 =4, a 3 =8, a 4 =9, etc. Then Conjecture : lim inf(a n a n )=. Results P. Mihăilescu, Catalan was right : the equation x p y q =where the unknowns x, y, p and q take integer values, all 2, has only one solution (x, y, p, q) =(3, 2, 2, 3). Previous partial results : J.W.S. Cassels, R. Tijdeman, M. Mignotte, / / 95

12 Higher values of k There is no value of k> for which one knows that Pillai s equation x p y q = k has only finitely many solutions. The abc Conjecture For a positive integer n, we denote by R(n) = Y p p n Pillai s conjecture as a consequence of the abc conjecture : the radical or the square free part of n. with x p y q c( )max{x p,y q } apple apple = p q Conjecture (abc Conjecture). For each " > 0 there exists apple(") such that, if a, b and c in Z >0 are relatively prime and satisfy a + b = c, then c<apple(")r(abc) +". 45 / / 95 The abc Conjecture of Œsterlé and Masser Shinichi Mochizuki The abc Conjecture resulted from a discussion between J. Œsterlé and D. W. Masser around 980. INTER-UNIVERSAL TEICHMÜLLER THEORY IV : LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS by Shinichi Mochizuki 47 / / 95

13 http :// motizuki/ Papers of Shinichi Mochizuki Shinichi General Arithmetic Geometry Intrinsic Hodge Theory p adic Teichmuller Theory Anabelian Geometry, the Geometry of Categories The Hodge-Arakelov Theory of Elliptic Curves Inter-universal Teichmuller Theory 49 / / 95 Shinichi Mochizuki [] Inter-universal Teichmuller Theory I : Construction of Hodge Theaters. PDF [2] Inter-universal Teichmuller Theory II : Hodge-Arakelov-theoretic Evaluation. PDF [3] Inter-universal Teichmuller Theory III : Canonical Splittings of the Log-theta-lattice. PDF [4] Inter-universal Teichmuller Theory IV : Log-volume Computations and Set-theoretic Foundations. PDF Beal Equation x p + y q = z r Assume and x, y, z are relatively prime p + q + r < Only 0 solutions (up to obvious symmetries) are known +2 3 =3 2, =3 4, =2 9, =7 2, =22 2, = , =65 7, =3 7, = , = / / 95

14 Beal Conjecture and prize problem Fermat-Catalan Conjecture (H. Darmon and A. Granville) : the set of solutions to x p + y q = z r with (/p)+(/q)+(/r) < is finite. Consequence of the abc Conjecture. Hint: p + q + r < implies p + q + r apple 4 42 Conjecture of R. Tijdeman, D. Zagier and A. Beal : there is no solution to x p + y q = z r where each of p, q and r is 3. R. D. Mauldin, A generalization of Fermat s last theorem : the Beal conjecture and prize problem, Notices Amer. Math. Soc., 44 (997), pp Example related to the abc conjecture =23 5 Continued fraction of 09 /5 : [2;,, 4, 77733,...], approximation : 23/9 09 /5 = = N. A. Carella. Note on the ABC Conjecture 53 / / 95 Waring s Problem In 770, a few months before J.L. Lagrange solved a conjecture of Bachet and Fermat by proving that every positive integer is the sum of at most four squares of integers, E. Waring wrote : Edward Waring ( ) Every integer is a cube or the sum of two, three,... nine cubes ; every integer is also the square of a square, or the sum of up to nineteen such ; and so forth. Similar laws may be a rmed for the correspondingly defined numbers of quantities of any like degree. Waring s function g(k) Waring s function g is defined as follows : For any integer k 2, g(k) is the least positive integer s such that any positive integer N can be written x k + + x k s. Conjecture (The ideal Waring s Theorem) :For each integer k 2, g(k) =2 k +[(3/2) k ] 2. This is true for 3 apple k apple , and (K. Mahler) also for all su ciently large k. 55 / / 95

15 n = x x 4 g : g(4) = 9 Waring s Problem and the abc Conjecture Any positive integer is the sum of at most 9 biquadrates R. Balasubramanian, J-M. Deshouillers, F. Dress (986). S. David : the estimate k k 3 3, 2 4 (for su ciently large k) followsnotonlyfrom Mahler s estimate,butalsofromtheabc Conjecture! Hence the ideal Waring Theorem g(k) =2 k +[(3/2) k ] 2. would follow from an explicit solution of the abc Conjecture. 57 / / 95 Waring s function G(k) G(2) = 4 Waring s function G is defined as follows : For any integer k 2, G(k) is the least positive integer s such that any su ciently large positive integer N can be written x k + + x k s. Joseph-Louis Lagrange (736 83) Solution of a conjecture of Bachet and Fermat in 770 : Every positive integer is the sum of at most four squares of integers, G(k) is known only in two cases : G(2) = 4 and G(4) = 6 No integer congruent to squares of integers. modulo 8 can be a sum of three 59 / / 95

16 G(k) Baker s explicit abc conjecture Kempner (92) G(4) 6 6 m 3 need at least 6 biquadrates Hardy Littlewood (920) G(4) apple 2 circle method, singular series Davenport, Heilbronn, Esterman (936) G(4) apple 7 Davenport (939) G(4) = 6 Yu. V. Linnik (943) g(3) = 9, G(3) apple 7 Other estimates for G(k), k 5 : Davenport, K. Sambasiva Rao, V. Narasimhamurti, K. Thanigasalam, R.C. Vaughan,... Alan Baker Shanta Laishram 6 / / 95 Real numbers : rational, irrational Rational numbers : a/b with a and b rational integers, b>0. Irreducible representation : p/q with p and q in Z, q>0 and gcd(p, q) =. Irrational number : a real number which is not rational. Complex numbers : algebraic, transcendental Algebraic number : a complex number which is a root of a non-zero polynomial with rational coe cients. Examples : rational numbers : a/b, rootofbx a. p 2,rootofX 2 2. i, rootofx 2 +. The sum and the product of algebraic numbers are algebraic numbers. The set of complex algebraic numbers is a field, the algebraic closure of Q in C. A transcendental number is a complex number which is not algebraic. 63 / / 95

17 Inverse Galois Problem Srinivasa Ramanujan A number field is a finite extension of Q. Is any finite group G the Galois group of a number field over Q? Evariste Galois (8 832) Equivalently : The absolute Galois group of the field Q is the group Gal(Q/Q) of automorphisms of the field Q of algebraic numbers. The previous question amounts to deciding whether any finite group G is a quotient of Gal(Q/Q). Some transcendental aspects of Ramanujan s work. Proceedings of the Ramanujan Centennial International Conference (Annamalainagar, 987), RMS Publ.,, Ramanujan Math. Soc., Annamalainagar, 988, / / 95 Periods : Maxime Kontsevich and Don Zagier Periods, Mathematics unlimited 200 and beyond, Springer 200, A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coe cients, over domains in R n given by polynomial inequalities with rational coe cients. The number Basic example of a period : = = Z Z e z+2i = e z Z 2i = Z x 2 +y 2 apple dx p x 2 z = dz z dxdy =2 = Z Z p dx x 2 x2 dx 67 / / 95

18 Further examples of periods p 2= Z and all algebraic numbers. Z log 2 = 2x 2 apple <x<2 dx dx x and all logarithms of algebraic numbers. Z = dxdy, x 2 +y 2 apple A product of periods is a period (subalgebra of C), but / is expected not to be a period. Relations among periods Additivity (in the integrand and in the domain of integration) Z b a f(x)+g(x) dx = Z b a f(x)dx = Z c a Z b a f(x)dx + f(x)dx + Z b c Z b a f(x)dx. g(x)dx, 2 Change of variables : if y = f(x) is an invertible change of variables, then Z f(b) f(a) F (y)dy = Z b a F f(x) f 0 (x)dx. 69 / / 95 Relations among periods (continued) Conjecture of Kontsevich and Zagier A widely-held belief, based on a judicious combination of experience, analogy, and wishful thinking, is the following 3 Newton Leibniz Stokes Formula Z b a f 0 (x)dx = f(b) f(a). Conjecture (Kontsevich Zagier). If a period has two integral representations, then one can pass from one formula to another by using only rules, 2, 3 in which all functions and domains of integration are algebraic with algebraic coe cients. 7 / / 95

19 Conjecture of Kontsevich and Zagier (continued) In other words, we do not expect any miraculous coïncidence of two integrals of algebraic functions which will not be possible to prove using three simple rules. This conjecture, which is similar in spirit to the Hodge conjecture, is one of the central conjectures about algebraic independence and transcendental numbers, and is related to many of the results and ideas of modern arithmetic algebraic geometry and the theory of motives. Conjectures by S. Schanuel, A. Grothendieck and Y. André Schanuel : if x,...,x n are Q linearly independent complex numbers, then n at least of the 2n numbers x,...,x n, e x,...,e xn are algebraically independent. Periods conjecture by Grothendieck : Dimension of the Mumford Tate group of a smooth projective variety. Y. André : generalization to motives. 73 / / 95 S. Ramanujan, C.L. Siegel, S. Lang, K. Ramachandra Ramanujan : Highly composite numbers. Alaoglu and Erdős (944), Siegel, Schneider, Lang, Ramachandra Four exponentials conjecture Let t be a positive real number. Assume 2 t and 3 t are both integers. Prove that t is an integer. Equivalently : If n is a positive integer such that (log 3)/ log 2 n is an integer, then n is a power of 2 : 2 k(log 3)/ log 2 =3 k. 75 / / 95

20 First decimals of p 2 First binary digits of p / / 95 Computation of decimals of p decimals computed by hand by Horace Uhler in decimals computed in decimals in decimals computed by Yasumasa Kanada and Daisuke Takahashi in 997 with Hitachi SR220 in 7 hours and 3 minutes. Émile Borel (87 956) Les probabilités dénombrables et leurs applications arithmétiques, Palermo Rend. 27, (909). Jahrbuch Database JFM Sur les chi res décimaux de p 2 et divers problèmes de probabilités en chaînes, C. R. Acad. Sci., Paris 230, (950). Zbl Motivation : computation of. 79 / / 95

21 Émile Borel : 950 Conjecture of Émile Borel Let g 2 be an integer and x arealirrationalalgebraic number. The expansion in base g of x should satisfy some of the laws which are valid for almost all real numbers (for Lebesgue s measure). Conjecture (É. Borel). Let x be an irrational algebraic real number, g 3 apositiveintegerandaan integer in the range 0 apple a apple g. Then the digit a occurs at least once in the g ary expansion of x. Corollary. Each given sequence of digits should occur infinitely often in the g ary expansion of any real irrational algebraic number. (consider powers of g). An irrational number with a regular expansion in some base g should be transcendental. 8 / / 95 The state of the art Complexity of the expansion in basis b of a real irrational algebraic number There is no explicitly known example of a triple (g, a, x), where g 3 is an integer, a adigitin{0,...,g } and x an algebraic irrational number, for which one can claim that the digit a occurs infinitely often in the g ary expansion of x. A stronger conjecture, also due to Borel, is that algebraic irrational real numbers are normal : each sequence of n digits in basis g should occur with the frequency /g n, for all g and all n. 83 / 95 Theorem (B. Adamczewski, Y. Bugeaud 2005 ; conjecture of A. Cobham 968). If the sequence of digits of a real number x is produced by a finite automaton, then x is either rational or else transcendental. 84 / 95

22 Open problems (irrationality) Catalan s constant Is the number e + = irrational? Is the number e = irrational? Is the number Is Catalan s constant X ( ) n (2n +) 2 n = an irrational number? log = irrational? 85 / / 95 Riemann zeta function The function (s) = X n s n was studied by Euler ( ) for integer values of s and by Riemann (859) for complex values of s. Euler : for any even integer value of s arationalmultipleof s. 2, the number (s) is Examples : (2) = 2 /6, (4) = 4 /90, (6) = 6 /945, (8) = 8 /9450 Introductio in analysin infinitorum Leonhard Euler ( ) Introductio in analysin infinitorum (748) Coe cients : Bernoulli numbers. 87 / / 95

23 Riemann zeta function Riemann zeta function The number (3) = X n =, n3 is irrational (Apéry 978). Recall that (s)/ s is rational for any even value of s 2. Is the number (5) = X n irrational? = n5 T. Rivoal (2000) : infinitely many (2n +)are irrational. Open question : Is the number (3)/ 3 irrational? 89 / / 95 Euler Mascheroni constant Euler s Constant is = lim + n! n Lorenzo Mascheroni ( ) log n = Is it a rational number? X = log + = k k = k= Z Z 0 0 ( x)dxdy ( xy)log(xy) Z [x] dx x Other open problems Artin s Conjecture (927) : given an integer a which is not a sqare nor, there are infinitely many p such that a is a primitive root modulo p. (+ Conjectural asymptotic estimate for the density). Lehmer s problem : Let 6= 0be an algebraic integer of degree d, and M( ) = Q d i= max(, i ), where = and 2,, d are the conjugates of. Is there is a constant c> such that M( ) <cimplies that is a root of unity? c< (Lehmer 933). Schinzel Hypothesis H. For instance : are there infinitely many primes of the form x 2 +? The Birch and Swinnerton Dyer Conjecture Langlands program 9 / / 95

24 Collatz equation (Syracuse Problem) The Kerala School of Mathematics (KSoM) February 4-8, 203, Calicut = Kozhikode Iterate n 7! Is (4, 2, ) the only cycle? ( n/2 if n is even, 3n + if n is odd. Workshop on number theory and dynamical systems. Pietro Corvaja, S.G. Dani, Michel Laurent, Michel Waldschmidt apply for the workshop at KSOM through online registration on or before 5th December / / 95 October 2, 202 Delhi University, South Campus Department of Mathematics Number theory: Challenges of the twenty first century Michel Waldschmidt Institut de Mathématiques de Jussieu Paris VI miw/ 95 / 95