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Subject: sci.math: Frequently Asked Questions [1/3] Newsgroups: sci.math,sci.answers,news.answers From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz) Date: Sun, 17 Apr 1994 20:00:40 GMT Archive-Name: sci-math-faq/part1 Last-modified: October 26, 1993 Version: 5.0 This is the list of Frequently Asked Questions for sci.math (version 5.0). Any contributions/suggestions/corrections are most welcome. Please use * e-mail * on any comment concerning the FAQ list. Section 1 of 3, questions 1Q to 5Q. Table of Contents ----------------- 1Q.- Fermat's Last Theorem, status of .. 2Q.- Values of Record Numbers 3Q.- Formula for prime numbers... 4Q.- Digits of Pi, computation and references 5Q.- Odd Perfect Number 6Q.- Computer Algebra Systems, application of .. 7Q.- Computer Algebra Systems, references to .. 8Q.- Fields Medal, general info .. 9Q.- Four Colour Theorem, proof of .. 10Q.- 0^0=1. A comprehensive approach 11Q.- 0.999... = 1. Properties of the real numbers .. 12Q.- There are three doors, The Monty Hall problem, Master Mind and other games .. 13Q.- Surface and Volume of the n-ball 14Q.- f(x)^f(x)=x, name of the function .. 15Q.- Projective plane of order 10 .. 16Q.- How to compute day of week of a given date 17Q.- Axiom of Choice and/or Continuum Hypothesis? 18Q.- Cutting a sphere into pieces of larger volume 19Q.- Pointers to Quaternions 20Q.- Erdos Number 21Q.- Why is there no Nobel in mathematics? 22Q.- General References and textbooks... 23Q.- Interest Rate... 24Q.- Euler's formula e^(i Pi) = - 1 ... 1Q: What is the current status of Fermat's last theorem? and Did Fermat prove this theorem? Fermat's Last Theorem: There are no positive integers x,y,z, and n > 2 such that x^n + y^n = z^n. I heard that <insert name here> claimed to have proved it but later on the proof was found to be wrong. ... A: The status of FLT has remained remarkably constant. Every few years, someone claims to have a proof ... but oh, wait, not quite. UPDATE... UPDATE... UPDATE Andrew Wiles, a researcher at Princeton, Cambridge claims to have found a proof. The proof was presented in Cambridge, UK during a three day seminar to an audience including some of the leading experts in the field. The manuscript has been submitted to INVENTIONES MATHEMATICAE, and is currently under review. Preprints are not available until the proof checks out. Wiles is giving a full seminar on the proof this spring. The proof is long and cumbersome, but here are some of the first few details: *From Ken Ribet: Here is a brief summary of what Wiles said in his three lectures. The method of Wiles borrows results and techniques from lots and lots of people. To mention a few: Mazur, Hida, Flach, Kolyvagin, yours truly, Wiles himself (older papers by Wiles), Rubin... The way he does it is roughly as follows. Start with a mod p representation of the Galois group of Q which is known to be modular. You want to prove that all its lifts with a certain property are modular. This means that the canonical map from Mazur's universal deformation ring to its "maximal Hecke algebra" quotient is an isomorphism. To prove a map like this is an isomorphism, you can give some sufficient conditions based on commutative algebra. Most notably, you have to bound the order of a cohomology group which looks like a Selmer group for Sym^2 of the representation attached to a modular form. The techniques for doing this come from Flach; you also have to use Euler systems a la Kolyvagin, except in some new geometric guise. CLARIFICATION: This step in Wiles' manuscript, the Selmer group bound, is currently considered to be incomplete by the reviewers. Yet the reviewers (or at least those who have gone public) have confidence that Wiles will fill it in. (Note that such gaps are quite common in long proofs. In this particular case, just such a bound was expected to be provable using Kolyvagin's techniques, independently of anyone thinking of modularity. In the worst of cases, and the gap is for real, what remains has to be recast, but it is still extremely important number theory breakthrough work.) If you take an elliptic curve over Q, you can look at the representation of Gal on the 3-division points of the curve. If you're lucky, this will be known to be modular, because of results of Jerry Tunnell (on base change). Thus, if you're lucky, the problem I described above can be solved (there are most definitely some hypotheses to check), and then the curve is modular. Basically, being lucky means that the image of the representation of Galois on 3-division points is GL(2,Z/3Z). Suppose that you are unlucky, i.e., that your curve E has a rational subgroup of order 3. Basically by inspection, you can prove that if it has a rational subgroup of order 5 as well, then it can't be semistable. (You look at the four non-cuspidal rational points of X_0(15).) So you can assume that E[5] is "nice." Then the idea is to find an E' with the same 5-division structure, for which E'[3] is modular. (Then E' is modular, so E'[5] = E[5] is modular.) You consider the modular curve X which parameterizes elliptic curves whose 5-division points look like E[5]. This is a "twist" of X(5). It's therefore of genus 0, and it has a rational point (namely, E), so it's a projective line. Over that you look at the irreducible covering which corresponds to some desired 3-division structure. You use Hilbert irreducibility and the Cebotarev density theorem (in some way that hasn't yet sunk in) to produce a non-cuspidal rational point of X over which the covering remains irreducible. You take E' to be the curve corresponding to this chosen rational point of X. *From the previous version of the FAQ: (b) conjectures arising from the study of elliptic curves and modular forms. -- The Taniyama-Weil-Shmimura conjecture. There is a very important and well known conjecture known as the Taniyama-Weil-Shimura conjecture that concerns elliptic curves. This conjecture has been shown by the work of Frey, Serre, Ribet, et. al. to imply FLT uniformly, not just asymptotically as with the ABC conj. The conjecture basically states that all elliptic curves can be parameterized in terms of modular forms. There is new work on the arithmetic of elliptic curves. Sha, the Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way an interesting aspect of this work is that there is a close connection between Sha, and some of the classical work on FLT. For example, there is a classical proof that uses infinite descent to prove FLT for n = 4. It can be shown that there is an elliptic curve associated with FLT and that for n=4, Sha is trivial. It can also be shown that in the cases where Sha is non-trivial, that infinite-descent arguments do not work; that in some sense 'Sha blocks the descent'. Somewhat more technically, Sha is an obstruction to the local-global principle [e.g. the Hasse-Minkowski theorem]. *From Karl Rubin: Theorem. If E is a semistable elliptic curve defined over Q, then E is modular. It has been known for some time, by work of Frey and Ribet, that Fermat follows from this. If u^q + v^q + w^q = 0, then Frey had the idea of looking at the (semistable) elliptic curve y^2 = x(x-a^q)(x+b^q). If this elliptic curve comes from a modular form, then the work of Ribet on Serre's conjecture shows that there would have to exist a modular form of weight 2 on Gamma_0(2). But there are no such forms. To prove the Theorem, start with an elliptic curve E, a prime p and let rho_p : Gal(Q^bar/Q) -> GL_2(Z/pZ) be the representation giving the action of Galois on the p-torsion E[p]. We wish to show that a _certain_ lift of this representation to GL_2(Z_p) (namely, the p-adic representation on the Tate module T_p(E)) is attached to a modular form. We will do this by using Mazur's theory of deformations, to show that _every_ lifting which 'looks modular' in a certain precise sense is attached to a modular form. Fix certain 'lifting data', such as the allowed ramification, specified local behavior at p, etc. for the lift. This defines a lifting problem, and Mazur proves that there is a universal lift, i.e. a local ring R and a representation into GL_2(R) such that every lift of the appropriate type factors through this one. Now suppose that rho_p is modular, i.e. there is _some_ lift of rho_p which is attached to a modular form. Then there is also a hecke ring T, which is the maximal quotient of R with the property that all _modular_ lifts factor through T. It is a conjecture of Mazur that R = T, and it would follow from this that _every_ lift of rho_p which 'looks modular' (in particular the one we are interested in) is attached to a modular form. Thus we need to know 2 things: (a) rho_p is modular (b) R = T. It was proved by Tunnell that rho_3 is modular for every elliptic curve. This is because PGL_2(Z/3Z) = S_4. So (a) will be satisfied if we take p=3. This is crucial. Wiles uses (a) to prove (b) under some restrictions on rho_p. Using (a) and some commutative algebra (using the fact that T is Gorenstein, 'basically due to Mazur') Wiles reduces the statement T = R to checking an inequality between the sizes of 2 groups. One of these is related to the Selmer group of the symmetric square of the given modular lifting of rho_p, and the other is related (by work of Hida) to an L-value. The required inequality, which everyone presumes is an instance of the Bloch-Kato conjecture, is what Wiles needs to verify. He does this using a Kolyvagin-type Euler system argument. This is the most technically difficult part of the proof, and is responsible for most of the length of the manuscript. He uses modular units to construct what he calls a 'geometric Euler system' of cohomology classes. The inspiration for his construction comes from work of Flach, who came up with what is essentially the 'bottom level' of this Euler system. But Wiles needed to go much farther than Flach did. In the end, _under_certain_hypotheses_ on rho_p he gets a workable Euler system and proves the desired inequality. Among other things, it is necessary that rho_p is irreducible. Suppose now that E is semistable. Case 1. rho_3 is irreducible. Take p=3. By Tunnell's theorem (a) above is true. Under these hypotheses the argument above works for rho_3, so we conclude that E is modular. Case 2. rho_3 is reducible. Take p=5. In this case rho_5 must be irreducible, or else E would correspond to a rational point on X_0(15). But X_0(15) has only 4 noncuspidal rational points, and these correspond to non-semistable curves. _If_ we knew that rho_5 were modular, then the computation above would apply and E would be modular. We will find a new semistable elliptic curve E' such that rho_{E,5} = rho_{E',5} and rho_{E',3} is irreducible. Then by Case I, E' is modular. Therefore rho_{E,5} = rho_{E',5} does have a modular lifting and we will be done. We need to construct such an E'. Let X denote the modular curve whose points correspond to pairs (A, C) where A is an elliptic curve and C is a subgroup of A isomorphic to the group scheme E[5]. (All such curves will have mod-5 representation equal to rho_E.) This X is genus 0, and has one rational point corresponding to E, so it has infinitely many. Now Wiles uses a Hilbert Irreducibility argument to show that not all rational points can be images of rational points on modular curves covering X, corresponding to degenerate level 3 structure (i.e. im(rho_3) not GL_2(Z/3)). In other words, an E' of the type we need exists. (To make sure E' is semistable, choose it 5-adically close to E. Then it is semistable at 5, and at other primes because rho_{E',5} = rho_{E,5}.) Referencesm: American Mathematical Monthly January 1994. Notices of the AMS, Februrary 1994. 2Q: What are the values of: largest known Mersenne prime? A: 2^859433-1 is prime. It was discovered in 1994. largest known prime? A: The largest known prime is the Mersenne prime described above. The largest known non-Mersenne prime, is 391581*2^216193-1. See Brown, Noll, Parady, Smith, Smith, and Zarantonello, Letter to the editor, American Mathematical Monthly, vol. 97, 1990, p. 214. Throughout history, the largest known prime has almost always been a Mersenne prime; the period between Brown et al's discovery in Aug 1989 and Slowinski & Gage's in March 1992 is one of the few exceptions. largest known twin primes? A: The largest known twin primes are 1691232 * 1001 * 10^4020 +- 1, which is a number with 4030 digits, found by H. Dubner. The second largest known twin primes are 4650828 * 1001 * 10^3429 +- 1. They were found by H. Dubner References: B. K. Parady and J. F. Smith and S. E. Zarantonello, Smith, Noll and Brown. Largest known twin primes, Mathematics of Computation, vol.55, 1990, pp. 381-382. largest Fermat number with known factorization? A: F_11 = (2^(2^11)) + 1 which was factored by Brent & Morain in 1988. F9 = (2^(2^9)) + 1 = 2^512 + 1 was factored by A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard in 1990. The factorization for F10 is NOT known. Are there good algorithms to factor a given integer? A: There are several that have subexponential estimated running time, to mention just a few: Continued fraction algorithm, Class group method, Quadratic sieve algorithm, Elliptic curve algorithm, Number field sieve, Dixon's random squares algorithm, Valle's two-thirds algorithm, Seysen's class group algorithm, A.K. Lenstra, H.W. Lenstra Jr., "Algorithms in Number Theory", in: J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, Elsevier, pp. 673-715, 1990. List of record numbers? A: Chris Caldwell maintains "THE LARGEST KNOWN PRIMES (ALL KNOWN PRIMES WITH 2000 OR MORE DIGITS)"-list. Send him mail to bf04@UTMartn.bitnet (preferred) or kvax@utkvx.UTK.edu, on any new gigantic primes (greater than 10,000 digits), titanic primes (greater than 1000 digits). What is the current status on Mersenne primes? A: Mersenne primes are primes of the form 2^p-1. For 2^p-1 to be prime we must have that p is prime. The following Mersenne primes are known. nr p year by ----------------------------------------------------------------- 1-5 2,3,5,7,13 in or before the middle ages 6-7 17,19 1588 Cataldi 8 31 1750 Euler 9 61 1883 Pervouchine 10 89 1911 Powers 11 107 1914 Powers 12 127 1876 Lucas 13-14 521,607 1952 Robinson 15-17 1279,2203,2281 1952 Lehmer 18 3217 1957 Riesel 19-20 4253,4423 1961 Hurwitz & Selfridge 21-23 9689,9941,11213 1963 Gillies 24 19937 1971 Tuckerman 25 21701 1978 Noll & Nickel 26 23209 1979 Noll 27 44497 1979 Slowinski & Nelson 28 86243 1982 Slowinski 29 110503 1988 Colquitt & Welsh jr. 30 132049 1983 Slowinski 31 216091 1985 Slowinski 32? 756839 1992 Slowinski & Gage 33? 859433 1993 Slowinski. The way to determine if 2^p-1 is prime is to use the Lucas-Lehmer test: Lucas_Lehmer_Test(p): u := 4 for i from 3 to p do u := u^2-2 mod 2^p-1 od if u == 0 then 2^p-1 is prime else 2^p-1 is composite fi The following ranges have been checked completely: 2 - 355K and 430K - 520K More on Mersenne primes and the Lucas-Lehmer test can be found in: G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, fifth edition, 1979, pp. 16, 223-225. 3Q.- Formula for prime numbers... Is there a polynomial which gives all the prime numbers? No, there is not. This is a simple exercise to prove. Is there a non-constant polynomial that only takes on prime values? It has been proved that no such polynomial exists. The proof is simple enough that an high school student could probably discover it. See, for example, Ribenboim's book _The Book of Prime Number Records_. Note, however, by the work of Jones, Sato, Wada, and Wiens, there *is* a polynomial in 26 variables such that the set of primes coincides with the set of *positive* values taken by this polynomial. See Ribenboim, pp. 147-150. But most people would object to the term "formula" restricted to mean polynomial. Can we not use summation signs, factorial, and the floor function in our "formula"? If so, then indeed, there *are* formulas for the prime numbers. Some of them are listed below. If we can't, then exactly what operations do you allow and why? Indeed, as I have previously argued, a reasonable interpretation of the word "formula" is simply "Turing machine that halts on all inputs". Under this interpretation, there certainly are halting Turing machines which compute the n'th prime number. However, nobody knows how to compute the n'th prime in time polynomial in log n. That's still an open question. Herb Wilf has addressed the question, "What is a formula?" in his delightful article, "What is an answer?" which appeared in the American Mathematical Monthly, 89 (1982), 289-292. He draws the distinction between "formula" and "good formula". Anyone who claims "there is no formula for the prime numbers" should read this article. Here are just a few articles that discuss "formulas" for primes. Almost all of these do *not* require computation of the primes "ahead of time". Most of them rely on standard mathematical functions such as summation, factorial, greatest integer function, etc. C. Isenkrahe, Math. Annalen 53 (1900), 42-44. W. H. Mills, Bull. Amer. Math. Soc. 53 (1947), 604. L. Moser, Math. Mag. 23 (1950), 163-164. E. M. Wright, Amer. Math. Monthly 58 (1951), 616-618. (Correction, 59 (1952), 99.) E. M. Wright, J. Lond. Math. Soc. 29 (1954), 63-71. B. R. Srinivasan, J. Indian Math. Soc. 25 (1961), 33-39. C. P. Willans, Math. Gazette 48 (1964), 413-415. V. C. Harris, Nordisk Mat. Tidskr. 17 (1969), 82. U. Dudley, Amer. Math. Monthly 76 (1969), 23-28. C. Vanden Eynden, Amer. Math. Monthly 79 (1972), 625. S. W. Golomb, Amer. Math. Monthly 81 (1974), 752-754. For more references see J.O. Shallit, E. Bach, _Algorithmic Number Theory_ (to be published, MIT Press). 4Q: Where I can get pi up to a few hundred thousand digits of pi? Does anyone have an algorithm to compute pi to those zillion decimal places? A: MAPLE or MATHEMATICA can give you 10,000 digits of Pi in a blink, and they can compute another 20,000-1,000,000 overnight (range depends on hardware platform). It is possible to retrieve 1.25+ million digits of pi via anonymous ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and pi.dat.Z which reside in subdirectory doc/misc/pi. New York's Chudnovsky brothers have computed 2 billion digits of pi on a homebrew computer. How is pi calculated to many decimals ? There are essentially 3 different methods. 1) One of the oldest is to use the power series expansion of atan(x) atan(x)=x-x^3/3+x^5/5-... together with formulas like pi=16*atan(1/5)-4*atan(1/239). This gives about 1.4 decimals per term. 2) A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book of Borwein and Borwein: Pi and the AGM, Canadian Math. Soc. Series, John Wiley and Sons, New York, 1987. They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program. 3) A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming): Set k1=545140134;k2=13591409;k3=640320;k4=100100025;k5=327843840;k6=53360; Then in AmsTeX notation $\pi=\frac{k6\sqrt(k3)}{S}$, where $$S=\sum_{n=0}^\infty (-1)^n\frac{(6n)!(k2+nk1)}{n!^3(3n)!(8k4k5)^n}$$ The great advantages of this formula are that 1) It converges linearly, but very fast (more than 14 decimal digits per term). 2) The way it is written, all operations to compute S can be programmed very simply since it only involves multiplication/division by single precision numbers. This is why the constant 8k4k5 appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals. Question: how can I get a C program which computes pi? Answer: if you are too lazy to use the hints above, you can use the following 160 character C program (reportedly by Dik T. Winter) which computes pi to 800 decimal digits. If you want more, it is easy to modify, but you have to understand how it works first. int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5; for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a, f[b]=d%--g,d/=g--,--b;d*=b);} References : (This is a short version for a more comprehensive list contact Juhana Kouhia at jk87377@cc.tut.fi) J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to Pi", American Mathematical Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220. P. Beckman A history of pi Golem Press, CO, 1971 (fourth edition 1977) J.M. Borwein and P.B. Borwein The arithmetic-geometric mean and fast computation of elementary functions SIAM Review, Vol. 26, 1984, pp. 351-366 J.M. Borwein and P.B. Borwein More quadratically converging algorithms for pi Mathematics of Computation, Vol. 46, 1986, pp. 247-253 J.M. Borwein and P.B. Borwein Pi and the AGM - a study in analytic number theory and computational complexity Wiley, New York, 1987 Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computation of pi Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231-244 David Chudnovsky and Gregory Chudnovsky The computation of classical constants, Columbia University, Proc. Natl. Acad. Sci. USA, Vol. 86, 1989. Y. Kanada and Y. Tamura Calculation of pi to 10,013,395 decimal places based on the Gauss-Legendre algorithm and Gauss arctangent relation Computer Centre, University of Tokyo, 1983 Morris Newman and Daniel Shanks On a sequence arising in series for pi Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp. 199-217 E. Salamin Computation of pi using arithmetic-geometric mean Mathematics of Computation, Vol. 30, 1976, pp. 565-570 D. Shanks and J.W. Wrench, Jr. Calculation of pi to 100,000 decimals Mathematics of Computation, Vol. 16, 1962, pp. 76-99 Daniel Shanks Dihedral quartic approximations and series for pi J. Number Theory, Vol. 14, 1982, pp.397-423 David Singmaster The legal values of pi The Mathematical Intelligencer, Vol. 7, No. 2, 1985 Stan Wagon Is pi normal? The Mathematical Intelligencer, Vol. 7, No. 3, 1985 J.W. Wrench, Jr. The evolution of extended decimal approximations to pi The Mathematics Teacher, Vol. 53, 1960, pp. 644-650 5Q: Does there exist a number that is perfect and odd? A given number is perfect if it is equal to the sum of all its proper divisors. This question was first posed by Euclid in ancient Greece. This question is still open. Euler proved that if N is an odd perfect number, then in the prime power decomposition of N, exactly one exponent is congruent to 1 mod 4 and all the other exponents are even. Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod 4. A sketch of the proof appears in Exercise 87, page 203 of Underwood Dudley's Elementary Number Theory, 2nd ed. It has been shown that there are no odd perfect numbers < 10^300. Copyright Notice Copyright (c) 1993 A. Lopez-Ortiz This FAQ is Copyright (C) 1994 by Alex Lopez-Ortiz. This text, in whole or in part, may not be sold in any medium, including, but not limited to electronic, CD-ROM, or published in print, without the explicit, written permission of Alex Lopez-Ortiz. -------------------------------------------------------------------------- Questions and Answers Edited and Compiled by: Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca Department of Computer Science University of Waterloo Waterloo, Ontario Canada -- Alex Lopez-Ortiz alopez-o@neumann.UWaterloo.ca Department of Computer Science University of Waterloo Waterloo, Ontario Canada