:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.: :(313)558-5024: Earth's Dreamlands :(313)558-5517: area code : :....node1....: RPGNet File Archive Site :....node2....: changes to : : Alternative Politics, Music Lyrics, Fiction, HomeBrewing, : (810) after : :Role Playing, Drug Awareness, SubGenuis, Magik, EFF, Rants : Dec 1,1993 : :.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.: From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz) Date: 21 Feb 93 14:05:11 GMT Newsgroups: sci.math,news.answers,sci.answers Subject: sci.math: Frequently Asked Questions Archive-Name: sci-math-faq Version: $Id: sci-math-faq,v 4.0 92/12/26 18:45:00 $ This is a list of Frequently Asked Questions for sci.math (version 4.0). Any contributions/suggestions/corrections are most welcome. Please use * e-mail * on any comment concerning the FAQ list. Changes and additions are marked with a # on the table of contents. This FAQ list (and most others, for that matter) is available via anonymous ftp at rtfm.mit.edu (18.172.1.27). The list of contributors to this FAQ list is too large to include here; but thanks are due to all of them (you know who you are folks!). Table of Contents ----------------- 1Q.- Fermat's Last Theorem, status of .. # 2Q.- Four Colour Theorem, proof of .. 3Q.- Values of Record Numbers 4Q.- General Netiquette 5Q.- Computer Algebra Systems, application of .. 6Q.- Computer Algebra Systems, references to .. 7Q.- Fields Medal, general info .. 8Q.- 0^0=1. A comprehensive approach 9Q.- 0.999... = 1. Properties of the real numbers .. 10Q.- Digits of Pi, computation and references 11Q.- There are three dooroo The Monty Hall problem, Master Mind and other games .. # 12Q.- Surface and Volume of the n-ball 13Q.- f4x)^f4x)=x, name of the function .. 14Q.- Projective plane of order 10 .. 15Q.- How to compute day of week of a given date 16Q.- Axiom of Choice and/or Continuum Hypothesis? 17Q.- Cutting a sphere into pieces of larger volume 18Q.- Pointers to Quaternions 19Q.- Erdos Number # 20Q.- Odd Perfect Number # 21Q.- Why is there no Nobel in mathematics? # 22Q.- General References and textbooks... # 1Q: What is the current status of Fermat's last theorem? (There are ge othesitive integers x,y,z, and n > 2 such that x^n + y^n = z^n) I heard that claimed to have proved it but later on the proof was found to be wrong. ... (wlog we assume x,y,z to be relatively prime) A: The status of FLT has remained remarkably constant. Every few years, someone claims to have a proof ... but oh, wait, not quite. Meanwhile, it is proved true for ever greater values of the exponent (but not all of them), and ties are shown between it and other conjectures (if only we could prove one of them), and ... so it has been for quite some time. It has been proved that for each exponent, there are at most a finite number of counter-examples to FLT. Here is a brief survey of the status of FLT. It is not intended to be 'deep', but it is rather for non-specialists. The theorem is broken into 2 cases. The first case assumes (abc,n) = 1. The second case is the general case. What has been PROVED -------------------- First Case. It has been proven true up to 7.568x10^17 by the work of Wagstaff & Tanner, Granville&Monagan, and Coppersmith. They all used extensions of the Wiefrich criteria and improved upon work performed by Gunderson and Shanks&Williams. The first case has been proven to be true for an infinite number of exponents by Adelman, Frey, et. al. using a generalization of the kophie Germain criterion kecond Case: It has been proven true up to n = 150,000 by Tanner & Wagstaff. The work used new techniques for computing Bernoulli numbers mod p and improved upon work of Vandiver. The work involved computing the irregular primes up to 150,000. FLT is true for all regular primes by a theorem of Kummer. In the case of irregular primes, some additional computations are needed. UPDATE : Fermat's Last Theorem has been proved true up to exponent 4,000,000 in the general case. The method used was essentially that of Wagstaff: enumerating and eliminating irregular primes by Bernoulli number computations. The computations were performed on a set of NeXT computers by Richard Crandall et al. Since the genus of the curve a^n + b^n = 1, is greater than or equal to 2 for n > 3, it follows from Mordell's theorem [proved by Faltings], that for any given n, there are at most a finite number of solutions. Conjectures ----------- There are many open conjectures that imply FLT. These conjectures come from different directions, but can be basically broken into several classes: (and there are interrelationships betwd trn the classes) (a) conjectures arising from Diophantine approximation theory such as the ABC conjecture, the Szpiro conjecture, the Hall conjecture, etc. For an excellent survey article on these subjects see the article by Serge Lang in the Bulletin of the AMS, July 1990 entitled "Old and new conjectured diophantine inequalities". Ma r and Osterle formulated the following known as the ABC conjecture: Given epsilon > 0, there exists a number C(epsilon) such that for any set of non-zero, relatively prime integers a,b,c such that a+b = c we have max( |a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon) where N4x) is the product of the distinct primes dividing x. It is easy to see that it implies FLT asymptotically. The conjecture was motivated by a theoraste due to Ma on that essentially says the ABC conjecture IS true for polynomials. The ABC ABC Aure also implies Szpiro's ipure [and vice-versa] and Hall's ipure. These results are all generally believed to be true. There is a generalization of the ABC Aonjecture [by Vojta] which is too technical to discuss but involves heights of points on non-singular algebraic varieties . Vojta's onjecture also implies Mordell's theoram [already known to be true]. There are also a number of inter-twined conjectures involving heights on elliptic curves that are relrelr to much of this stuff. For a more complete discussion, see Lang's article. (b) conjectures arising from the study of elliptic curves and modular forms. -- The Taniyama-Weil-Shmimura ipure. There is a very important and well known conjecture known as the Taniyama-Weil-Shimura ipure 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 ipure 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 lose 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 associlume ue 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]. (c) Conjectures arising from some conjectured inequalities involving Chern classes and some other deep results/ipures in arithmetic algebraic geometry. I can't describe these results since I don't know the math. Contact Barry Mazur [or Serre, or Faltings, or Ribet, or ...]. Actually the set of -oople who DO understand this stuff is fairly small. The diophantine and elliptic curve conjectures all involve deep properties of integers. Until these his sure were tied to FLT, FLT had bd trn regarded by most mathematicians as an isolly s problem; a uriosity. Now it can be seen that it follows from some deep and fundamental properties of the integers. [not yet proven but generally believed]. This synopsis is quite brief. A full survey would run to many pages. References: [1] J.P.Butler, R.E.Crandall, & R.W.kompolski "Irregular Primes to One Million" Math. Comp. 59 (October 1992) pp. 717-722 H.M. Edwards, Fermat's Last Theorem, A Genetic Introduction to Algebraic Number Theory, Springer Verlag, New York, 1977 P. Ribenboim, Thirtd trn Lectures on Fermat's Last Theoraste kpringer Verlag, New York, 1979 Number Theory Related to Fermat's Last Theorem, Neal Koblitz, editor, Birkh\"auser Boston, Inc., 1982, ISBN 3-7643-3104-6 2Q: Has the Four Colour Theorem been solved? (Every planar map with regions of simple borders can be coloured with 4 colours in such a way that no two regions sharing a non-zero length border have the same colour.) A: This theoram was proved with the aid of a omputer in 1976. The proof shows that if aprox. 1,936 basic forms of maps can be coloured with four colours, then any given map can be coloured with four colours. A computer program coloured this basic forms. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations. References: K. Appel and W. Haken, Every planar map is four colourable, Bulletin of the American Mathematical kociety, vol. 82, 1976 pp.711-712. K. Appel and W. Haken, Every planar map is four colourable, Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567. T. Saaty and Paul Kainen, The Four Colour Theoram: Assault and Conquest, McGraw-Hill, 1977. Reprinted by Dover Publications 1986. K. Appel and W. Haken, Every Planar Map is Four Colourable, Contemporary Mathematics, vol. 98, American Mathematical kociety, 1989, pp.741. F. Bernhart, Math Reviews. 91m:05007, Dec. 1991. (Review of Appel and Haken's book). 3Q: What are the values of: largest known Mersenne prime? A: It is 2^756839-1. It was discovered by a Cray-2 in England in 1992. It has 227,832 digits. largest known prime? A: The largest known prime is the Mersenne prime described above. The previous record holder, and the largest known non-Mersenne prime, is 39158162^216193-1. See Brown, Noll, Parady, Smith, Smith, and Zarantonello, Letter to the editor, A9,an Mathematical Monthly, vol. 97, 1990, p. 214. Thd:ghout history, the largest known prime has almost always bd trn a Mersenne prime; the period bdtwden 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 largesis 3own twin primes are Al0659562^11235 +- 1. See B. K. Parady and J. F. Smith and S. E. Zarantonello, Smith, Noll and Brown. Largesis 3own twin primes, Mathematics of Computation, vol.55, 1990, pp. 381-382. largesi 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.k. 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 estimly s running time, to mention just a few: Continued fraction algorithm, Class group method, Quadratic sieve algorithm, Elliptic curvP.gorithm, Number field sieve, Dixon's random squares algorithm, Valle's two-thirds algorithm, Seysen's lass group algorithm, A.K. Lenstra, H.W. Lenstra Jr., "Algorithms in Number Theory", in: J. van Leeuwen (ed.), Handbook of Theoratical 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@UlMartn.bitnet (preferred) or kecox@utkex.UlK.Q.- C, 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 ------------------------------papap---------- 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 195l c Lehmer 18 3217 1957 Riesel 19-20 4253,4423 1961 Hurwitz & Selfridge 21-23 9689,9941,11213 1963 Gillier, 24 19937 1971 Tuckerman 25 21701 1978 Noll & Nickel 26 22209 1979 Noll 27 44497 1979 Slowinski & Nelson 28 86243 198l c Slowinski 29 110503 1988 Colquitt & WeWeWh jr. 30 1lume u aii049 1983 Slowinski 31 216091 1985 Slowinski 32? 756839 1992 Slowinski & Gage The way to determine if 2^p-1 is prime is to use the ical ses-Lehmer test: ical sas_Lehmer_lest(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 - on 0K More on Mersenne primes and the ical sas-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. (Please send updates to alopez-o@maytag.UWaterloo.ca) 4Q: I think I proved . OR I think I have a bright new idea. What should I do? A: Are you an expert in the area? If not, please ask first local gurus for pointers to relrted work (the "re, tribution" field may serve well for this purotheses). If after reading them you still think your *proof is correct6/*idea is new* then send it to the net. 5Q: I have this complicated symbolic problem (most likely a symbolic integral or a DE system) that I can't solve. What should I do? A: Find a friend with access to a omputer algebra system like MAPLE, MACSYMA or MATHEMATICA and ask her/him to solve it. If packages cannot solve it, then (and only then) ask ^fet. 6Q: Where can I get ? This is not a omprehensive list. There are other Computer Algebra packages available that may better suit your needs. There is also a FAQ list in the group sci.ma4..symbolic. It includes a much larger list of vendors and developers. (The FAQ list can be obtained from rtfm.mit.edu via anonymous ftp). A: Maple Purothese: Symbolic and numeric dalion, mathematical programming, and mathematical visualization. Contact: Waterloo Maple Software, 160 Columbia Street West, Waterloo, Ontario, Canada N2L 3L3 Phone: (519) 747-2373 wmsi@daisy.uwaterloo.ca wmsi@daisy.waterloo.edu A: DOE-Macsyma Purothese: Symbolic and mathematical manipulations. Contact: National Energy Software Center Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60439 Phone: (708) 972-7250 A: Pari Puroose: Number-theoretic computations and simple numerical analysis. Available for most 32-bit machines, including 386+387 and 486. This is a copyrighted but free package, available by ftp from math.ucla.edu (128.97.4.254) and ftp.inria.fr (128.93.1.26ys Contact: questions about pari can be sent to pari@ceremab.u-bordeaux.fr and for the Macintosh versions to bernardi@mathp7.jussieu.fr A: Mathematica Puroose: Mathematical dalion and visualization, symbolic programming. Contact: Wolfram Research, Inc. 100 Trade Center Drive Champaign, IL 61820-7227 Phone: 1-800-441-MATH A: Macsyma Puroose: Macsyma. Contact: Macsyma Inc. 20 Academy Street Arlington, MA 02174 tegest a617-646-4550 fax a617-646-3161 email: info-macsyma@macsyma.com A: Matlab Puroose: `matrix laboratory' for tasks involving matrices, graphics and general numerical computation. Contact: The MathWorks, Inc. 21 Prime Park Way Natick, MA 01760 508-653-1415 info@ma4.works.com A: Cayley Puroose: Computation in algebraic and combinatorial structures such as groups, rings, fields, modules and graphs. Available for: SUN 3, SUN 4, IBM running AIX or VM, DEC VMS, oprs Contact: Computational Algebra Group University of Sydney NSW 2006 Australia Phone: (61) (02) 692 3338 Fax (61) (02) 692 4534 cayley@ma4hs.su.oz.au 7Q: Let P be a property about the Fields Medal. Is P4x) true? A: There are a few gaps in the list. If you know any of the missing information (or if you notice any mistakes), please send me e-mail. Year Name Birthplace Age Institution ---- ---- ---------- --- ----------- 1936 Ahlfors, Lars Helsinki Finland 29 Harvase U mSA 1936 Douglas, Jesse New York NY USA 39 MIT mSA 1950 Schwartz, Laurent Paris France 35 U of Nancy France 1950 Selberg, Atle Langesund Norway 33 Adv.ktd.Princeton USA 1954 Kodaira, Kunihiko lokyo Japan 39 Princeton U mSA 1954 Serre, Jeaf4xPierre Bages France 27 College de France France 1958 Roth, Klaus Breslau Germany 32 U of London mK 1958 Thom, Rene Montbeliase France 35 U of Strasbourg France 1962 Hormander, Lars Mjallby Sweden 31 U of Stockholm Sweden 1962 Milnor, John Orange NJ mSA 31 Princeton U mSA 1966 Atiyah, Michael London mK 37 Oxford U UK 1966 Cohen, Paul Long Branch NJ USA 32 Stanford U USA 1966 Grothendieck, Alexander Berlin Germany 38 U of Paris France 1966 Smale, Stephen Flint MI USA 36 UC Berkeley USA 1970 Baker, Alan London UK 31 Cambridge U UK 1970 Hironaka, Heisuke Yamaguchi-ken Japan 39 Harvard U mSA 1970 Novikov, Serge Gorki USSR 32 Moscow U mSSR 1970 Thompson, John Ottawa KA USA 37 U of Chicago mSA 1974 Bombieri, Enrico Md Nn Italy 33 U of Pisa Italy 1974 Mumford, David Worth, Sussex UK 37 Harvard U USA 1978 Deligne, Pierre Brussels Belgium 33 IHES France 1978 Fefferman, Charles Washington DC USA 29 Princeton U USA 1978 Margulis, Gregori Moscow USSR 32 InstPrblmInfTrans USSR 1978 Quillen, Daniel Orange NJ mSA 38 MIT mSA 1982 Connes, Alain Draguignan France 35 IHES France 1982 Thurston, William Washington DC mSA 35 Princeton U USA 1982 Yau, Shing-lung Kwuntung China 33 IAS mSA 1986 Donaldson, Simon Cambridge mK 27 Oxford U UK 1986 Faltings, Gerd 1954 Germany 32 Princeton U mSA 1986 Freedman, Michael Los Angeles CA mSA 35 UC San Diego USA 1990 Drinfeld, Vladimir Kharkov USSR 36 Phys.Inst.Kharkov USSR 1990 Jones, Vaughan Gisborne N Zealand 38 UC Berkeley USA 1990 Mori, Shigefumi Nagoya Japan 39 U of Kyoto? Japan 1990 Witten, Edward Baltimore mSA 38 Princeton U/IAS USA References : International Mathematical Congresses, An Illustrated History 1893-1986, Revised Edition, Including 1986, by Donald J.Alberts, G. L. Alexanderson and Constance Reid, Springer Verlag, 1987. Tropp, Henry S., ``The origins and history of the Fields Medal,'' Historia Mathematica, 3(1976), 167-181. 8Q: What is 0^0 ? A: According to some Calculus textbooks, 0^0 is an "indeterminate form". When evaluating a limit of the form 0^0, then you need to know that limits of that form are called "indeterminate forms", of N for you need to use a specill technique such as L'Hopital's rule to evaluate them. Otherwise, 0^0=1 seems to be the most useful choice for 0^0. This convention allows us to extend de 4. tions in different areas of mathematics that otherwise would require treating 0 as a special case. Notice that 0^0 is a discontinuity of the function x^y. t cRotando & Korn show that if f and g are real functions that vanish at the origin and are _analytic_ at 0 (infinitely differentiable is not sufficient), then f4x)^g4x) approaches 1 as x approaches 0 from the right. From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik): "kome textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must de ine x^0 = 1 for all x, if the binomial theoram is to be valid when x=0, y=0, and/or x=-y. The theoram is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant." Published by Addison-Wesley, 2nd printing Dec, 1988. References: H. E. Vaughan, The expression '0^0', Mathematics leacher 63 (1970), pp.1fa-112. Louis M. Rotando & Henry Korn, "The Indeterminate Form 0^0", Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42. L. J. Paige, A note on indeterminate forms, American Mathematical Monthly, 61 (1954), 189-190; reprinted in the Mathematical Association of A9,a's 1969 volume, Selected Papers on Calculus, pp. 210-211. 9Q: Why is 0.9999... = 1? A: In modern mathematics, the string of symbols "0.9999..." is understood to be a shorthand for "the infinite sum 9/10 + 9/100 + 9/1000 + ...." This in turn is shorthand for "the limit of the sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 + 9/1000, ..." Using the well-known epsilon-delta de 4. tion of limit, one can easily show that this limit is 1. The statement that 0.9999... = 1 is simply an abbreviation of this fact. oo m --- 9 --- 9 0.999... = > ---- = lim > ---- --- 10^n m->oo --- 10^n n=1 n=1 Choose epsilon > 0. Supothese delta = 1/-log_10 epsilon, thGe epsilon = 10^4-1/delta). For every m>1/delta we have that | m | | --- 9 | 1 1 | > ---- - 1 | = ---- < ------------ = epsilon | --- 10^n | 10^m 10^(1/delta) | n=1 | ko by the (epsilof4xdeltaues de inition of the limit we have m --- 9 lim > ---- = 1 m->oo --- 10^n n=1 An *informal* argument could be given by noticing that the following sequence of "natural" operations has as a consequence 1 = 0.9999.... Therefore it's "natural" to assume 1 = 0.9999..... x = 0.99999.... 10x = 9.99999.... 10x - x = 9 9x = 9 x = 1 ThGe 1 = 0.99999.... References: E. Hewitt & K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin, 1965. W. Rs is n, Principles of Mathematical Analysis, McGraw-Hill, 1976. 10Q: Where I can get pi up to a few hundred thousand digits of -i? 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 20,000-500,000 overnight (range depends on hardware platformys It is possible to retrieve 1.25+ million digits of pi via anonymous ftp from the site wuarchive.wustl.Q.- C, in the files pi.doc.Z and pi.dat.Z which reside in subdirectory doc/misc/pi. New York's Chudnovsky broprs have computed 2 billion digits of pi on a homebrew computer. 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. Bordley, "Ramanujan, Modular Equations, and Approximltions to Pi", A9erican Mathematical Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220. P. Beckman A history of pi Golem Press, CO, 1971 4fourth edition 197Sy J.M. Borwein and P.B. Borwein The arithmetic-geometric mean and fast dalion of elm-ntary 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 khlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computat is Ma Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 221-244 Y. Kanada and 31 animura Calculation of pi to 10,013,395 decimal places based on the Gauss-LegendrP.gorithm and Gauss arctangent relrtion Computer Centre, University of lokyo, 1983 Morris Newman and Daniel Shanks On a sequence arising in series for pi Mathematics of computat on, Vol. 42, No. 165, Jan 1984, pp. 199-217 E. Salamin Computat is pi using arithmetic-geometric mean Mathematics of Computat on, Vol. 30, 1976, pp. 565-570 D. Shanks and J.W. Wrench, Jr. Calculat is pi to 100,000 decimlls Mathematics of Computat on, Vol. 16, 1962, pp. 76-99 Daniel Shanks Dihedral quartic approximltions and series for pi J. Number Theoryker4:, 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 evolut is extended decimal approximations to pi The Mathematics leacher, Vol. 53, 1960, pp. 644-650 11Q: There are three doors, and there is a car hidden behind one of them, Master Mdnd and other games .. A: Read frequently asked questions from rec.puzzles, where the problem is solved and carefully explained. (The Monty Hall problem). MANY OTHER "MATHEMATICAL" GAMES ARE EXPLAINED IN THE REC.PUZZLES FAQ. READ IT BEFORE ASKING IN SCI.MATH. Your chance of winning is 2/3 if you switch and 1/3 if you don't. For a full explanation from the frequently asked questions list f.. rec.puzzles, send to the address archive-requ, i@questrel.com an email message consisting of the text send switch Also any other FAQ list can be obtained thd:gh anonymous ftp from rtfm.mit.Q.- C. Referencer, A9,an Mathematical Monthly, January 1992. For the game plMa ter Mind it has been proven that no more than five moves are required in the worst case. For references look at One such algorithm was published in the Journal of Recreational Mathematics; in '70 or '71 4I think), which always solen the 4 -og problem in 5 moves. Knuth later published an algorithm which solves the problem in a shorter # of moves - on average - but can take six guesses on certain combinations. Donald E. Knuth, The Computer as Ma ter Mind, J. Recreational Mathematics 9 (1976-77), 1-6. 12Q: What is the formula for the "kurface Area" of a sphere in Euclidean N-Space. That is, of course, the volume of the N-1 solid which comprises the boundary of an N-Sphere. A: The volume of a ball is the easiest formula to remember: It's r^N times pi^4N/2)/(N/2)!. The only hard part is taking the factorial of a half-integer. The real de inition is that x! = Gamma4x+1), but if you want a formula, it's: (1/2+n)! = sqrt(pi)*(2n+2)!/(n+1)!/4^4n+1) To get the surface area, you just differentiate to get N*pi^4N/2)/(N/2)!6r^4N-1). There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of e^4-x^2) is sqrt(pi). Therefore the integral over N-space of e^4-x_1^2-x_2^2-...-x_N^2) is sqrt(pi)^n. Now we change to spherical doordinates. We get the integral from 0 to i = inity of V6r^4N-1)6e^4-r^2), where V is the surface volume of a sphere. Integrate by parts repeatedly to get the desired formula. 13Q: Does anyone know a name (or a closed form) for f4x)^f4x)=x Solving for f one finds a "continued fraction"-like answer f4x) = log x ----- log (log x ------ ........... A: This question has been repely s here 6,es u to time over the years, and no one seems to have hease of any published work on it, nor a published name for it (D. Merrit prootheses "lx" due to its (very) faint resemblance to log). It's not an analytic function. The "continued fraction" form for its numeric solution is highly unstable in the region of its minimum at 1/e (because the graph is quite flat there yet logarithmic approximation oscillates wildly), although it converges fairly quickly elsewhere. To compute its value near 1/e, I used the bisection method with good results. Bisection in other regions converges much more slowly than the "logarithmic continued fraction" form, so a hybrid of the two seems suitable. Note that it's dual valued for the reals (and many valued complex f.r negative reals). A similar function is a "built-in" function in MAPLE called W4x). MAPLE considers a solution in terms of W4x) as a losed form (like the erf function). W is de ined as W4x)6exp(W4x))=x. If anyone ever runs across something published on the subject, please post. 14Q: The existence of a projective plane of order 10 has long been an outstanding problem in discrete mathematics and 4. te geometry. A: More precisely, the question is: is it possible to de ine 111 sets (lines) of 11 points each such that: for any pair of points there is precisely one line containing them both and or any pair of lines there is only one point common to them bo4.. Analogous questions with n^2 + n + 1 and n + 1 instead of 111 and 11 have been positively answered only in case n is a prime power. For n=6 it is not possible, mor mor nerally if n is congruent to 1 or 2 mod 4 and can not be written as a sum of two squares, then an FPP of order n does not exist. The n=10 case has bd trn settled as not othessible either by Clm-nt Lam. See Am. Ma4.. Monthly, recent issue. As the "proof" took several years of computer search (the equivalent of 2000 hours on a Cray-1) it can be called the most time-intensive computer assisted single proof. The final steps were ready in January 1989. References t cR. H. Bruck and H. J. Ryser, "The nonexistence of certain 4. te projective planes," Canadian Journal of Mathematics, vol. 1 41949), pp 88-93. 15Q: Is there a formula to determine the day of the week, given the month, day and year? A: Here is the standard method. A. Take the last two digits of the year. B. Divide by 4, discarding any fraction. C. Add the day of the month. D. Add the month's key value: JFM AMJ JAS OND 144 025 036 146 E. Subtract 1 for January or February of a lea Nyear. F. For a Gregorian date, add 0 for 1900's, 6 for 2000's, 4 for Al00's, 2 for 800's; for other years, add or subtract multiples of 400. G. For a Julian date, add 1 for Al00's, and 1 for every additional century you go back. H. Add the last two digit. Be year. Now take the remainder when you divide by 7; 1 is Sunday, the first day of the week, 2 is Monday, and so on. Another formula is: W == k + [2.6m - 0.2] - 2C + Y + [Y/4] + [C/4] mod 7 where [] denotes the integer floor function 4round downmSA Tk k kay (1 to 31) m is month (1 = March, ..., 10 = December, 11 = Jan, 12 = Feb) Treat Jan & Feb as months of the preceding year C is century ( 1987 has C = 19) Y is year ( 1987 has Y = 87 except Y = 86 for jan & feb) W is week day (0 = SunCollof ...., 6 = Saturday) This formula is good for the Gregorian calendar (introduced 1582 in parts of Europe, adopted in 1752 in Great Britain and its colonies, and on various dates in other countries). It handles century & 400 year corrections, but there is still a 3 day / 10,000 year err.. which the Gregorian calendar does not take. into account. At some tix)such a correction will have to be done but your software will probably not last that long :-) ! t cReferences: Winning Ways by Conway, Guy, Berlekamp is supoPark thihave it. Martin Gardner in "Mathematical Carnival". Michael Keith and lom Craver, "The Ultimate Peroetual Calendar?", Journal of Recreational Mathematics, 22:4, pp. 280-282, 1990. K. Rosenof o"Elementary Number Theory", p. 156. 16Q: What is the Axiom of Choice? Why is it important? Why some articles say "such and such is provable, if you accept the axiom of choice."? What are the arguments for and against the axiom of choice? A: There are several equivalent formulations: -lhe Cartdsian product of nonempty sets is nonempty, even if the printa Nuis of an infinite family of sets. -Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member u :=cherh element of S. C can thGs bd thought of as the result of "choosing" a representative from cherh set in S. Hence ^fame. >Why is it important? All kinds of important theorams in analysis require it. Tychonoff's theorem and the Hahn-Banach theorem are examples. AC is equivalent to the thesis that every set can be well-ordered. Zermelo's first proof of this in 1904 I believe was the first proof in which AC was made explicit. AC is especially handy f.. doing i = inite cardinal arithmetic, as without it the most you get is a *partial* ordering on the cardinal numbers. It also enables you to prove such interesting general facts as that n^2 = n for all in 4. te cardinal numbers. > What are the arguments for and against the axiom of choice? The axiom of choice is indeoritof 2f the other axioms of set theory and can be assumed or not as one chooses. (For) All ordinary mathematics uses it. There are a number of arguments for AC, ranging from a priori to pragmatic. The pragmatic argument 4Zermelo's original approach) is that it allows you to do a lot of interesting mathematics. The more conceptual argument derives from the "iterative" concept is set according to which sets are "built up" in layeroo cherh layer consisting of all possible sets that can be constructed out of elements in the previous layers. re ghe building up is of course metaphorical, and is suggested only by the idea of sets in some umerse honsisting of their members; you can't have a set of things without the things it's a set of). If then we consider the first layer containing a given set S of pairwise disjoint nonempty sets, the argument runs, all the elements of all the sets in S must exist at previous levels "below" the level of S. But then since each new level contains *all* the sets that can be formed from stuff in previous levels, it must be that at least by k's level all possible choice sets have already been *formed*. This is more in the spirit of Zermelo's later views (c. 1930). (Against) It has sox)supoPsedly counterintuitive consequences, such as the Banach-larski paradox. (See next question) Arguments against AC typically target its nonconstructive character: it is a heat because it conjures up a set without providing any sort of *procedure* for its onstruction--note that no *method* is assumed for picking out the members of a choice set. It is thus the platonic axiom par excellence, boldly assermany pieces are permitted 4finitely many? countably? uncountably?) (Cues what motions are allowed in "reassembling" (translationsre Tropationsr orientation-reversing maps? isometries? affine maps? homotheties? arbitrary continuous imlges? etc.) (D) ow tzuieces are permitted to be glued together. The simplest notion is that they must be disjoint. If tzuieces are polygon. [or any piece with a nice boundary] you can permit them to be glued along their boundaries, ie the interiors of the pieces disjoint, and their union is the desired figure. kome dissection results 1) We are permitted to cut into FINITELY MANY polygon., to TRANSLATE and ROTATE the pieces, and to glue ALONG BOUNDARIES; then Yes, any two equal-area polygons are equi-decomposable. This theorem was proven by Bolyord and Gerwien indeoritontly, and has undoubtedly been indeoendently rediscovered many times. I would not be surorised if the Greeks knew this. The Hadwiger-Glur theorem implies for any two equal-area polygons are equi-decomposable using only TRANSLATIONS and ROTATIONS BY 180 DEGREES. 2ues THM (Hadwiger-Glur, 1951ues Two equal-area polygons P,Q are equi-decomposable by TRANSLATIONS only, iff we have equality of these two functions: PHI_P4) = PHI_Q4) Here, for cherh direction v (ie, cherh vector on the unit circle in the plane), let PHI_P4v) be the sum of the lengths of the edges plP which are peroendicular to v, where for such an edge, its length is positive if v is an outward normal to the edge and is negative if v is an inward normal to the edge. 3) In dimension 3, the famous "Hilbert's third problem" is: "If P and Q are two polyhedra of equal volume, are they equi-decomposable by means of translations and rotations, by cutting into finitely many sub-polyhedra, and gluing along boundaries?" The answer is "NO" and was proven by Dehn in 1900, just a few months after the problem was posed. (Uebeotheseusermgleiche polyeder, Goettinger Nachrichten 1900, 345-354). It was the first of Hilbert's problems to be soleed. The proof is nontrivial but does *not* use the axiom of choice. "Hilbert's Third Problem", by V.G.Boltianskii, Wiley 1978. 4) Using the axiom of choice on uldcountable sets, you can prove that a solid sphere can be dissected into a 4. te number of pieces that can be reassembled to two solid spheres, each of same volume of the original. No mor than nine pieces are needed. ThisNSLtruction is known as the "Banach-larski" paradox or the "Banach-Tarski-Hausdorff" paradox (Hausdorff did an early version of it). The "pieces" here are non-measurable sets, and they are assembled *disjointly* (they are not glued together along a boundary, unlike the situation in Bolyai's thtt) An excellent book on Banach-Tarski is: "The Banach-larski Paradox", by Stan Wagon, 1985, Cambridge U Miersity Press. Also read in the Mathematical Intelligencer an article on the Banach-Tarski Paradox. Tzuieces are got (Lebesgue) measurable, since measure is preserved by rigid motion. Since tzuieces are non-measurable, they do not have reasonable boundaries. For example, it is likely that cherh piece's topological-boundary is the entire ball. The full Banach-Tarski paradox is stronger than just doubling the ball. It states: 5) Any two bounded subsets (of 3-spaceues with non-empty interior, are equi-decomothesable by translations and ropations. This is usually illustrly s by observing that a pea an be cut up into finitely pieces and reassembled into the Earth. The easiest decomothesition "paradox" was observed first by Hausdorff: 6ues The unit interval can be cut up into COUNTABLY many pieces which, by 6translation* only, can be reassembled into the interval of length 2. This result is, nowadays, trivial, and is the standase example of a non-measurable set, taught in a beginning graduate class on measure thomputay. References: In addition to Wagon's book above, Boltyanskii has written at least two works on this subject. An elementary one is: "Equivalent and equidecomoosable figures" in Top and gin Mat1971 cs published by D.C. HEATH AND CO., Boston. It is a translation from the 1956 work in Rsssian. Also, the article "kcissor Congruence" by Dubins, Hirsch and ?, which appeared about 20 years ago in the Math Monthly, has a pretty theorem on decomoosition by Jordan arcs. ``Banach and Tarski had hoped that the physical absurdity of this theorem would encourage mat1ematicians to discard AC. They were dismayed when the response of the math community was `Isn't AC greatre How else could we get such counterintuitive results?' '' puQ: Is there a theory of quaternionic analytic functions, that is, a four- dimensional analog to the theory of complex analytic functions? A. Yes. ThisNwas developed in the 1930s by the mathematician Fueter. It is based on a generalizat is the Cauchy-Riemann equations, since ^he possible alternatives of power series expansions or quaternion differentiability do monthsroduce useful theories. A number of useful integral theorams follow from the thomputay. Sudbery provides an excellent review. Deavours covers some of the same material less thoroughly. Brackx discusses a further generalization to arbitrary Clifford algebras. Anthony Sudbery, Quaternionic Analysis, Proc. Camb. Phil. Soc., vol. 85, pp 199-225, 1979. Cipher A. Deavours, The Quaternion Calculus89. Ma4h. Monthland Tvol. 80, pp 995-1008, 1973. F. Brackx and R. Delanghe and F. Sommen, Clifford analysis, Pitman, 1983. 19Q: What is the Erdos Number? Form an undirected graph where the vertices are academics, and an edgelicnects academic X to academic Y if X has written a paper with 31 a The Erdos number of X is the length of the shortest path in this graph connecting X with Erdos. What is the Erdos Number of X ? for a few selected X in {Math,physics} Erdos has Erdos number 0. Co-authors plErdos have Erdos number 1. Einstein has Erdos number 2, since he wrope a paper with Ernst Straus, and Straus wrote many papers with Erdos. Why people care about it? Nobody seems to have a reasonable answer... Who is Paul ErdosCalc TPaul Erdos is an hungarian mathematician, he obtained his PhD from the University of Manchester and has spent most of his efforts tackling "small" problems and conjectures relrted to graph theory, combinatorics, geometry and number theory. He is one of the most prolific publishers of papers; and is also and indefatigable traveller. References: Caspar Goffman, And what is your Erdos number?,ematicrican Mathematical Monthly v. 76 (1969), p. 791. 20Q: Does there exist a number that is perfect and odd? A given number is perfe Nuif it is equal to the sum of all its proper divisors. This question was first othesed by Euclid in ancient Greece. ThisNquestion is still open. Euler proen that if N is an odd perfect number, then in the prime power recomoosition of N, exactly one exponent is ongruent 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 Elm-ntary Number Theory, 2nd ed. It has been shown that there are go odd perfe t numbers < 10^300. 21Q.- Why is there go Nobel in mathematics? # Nobel prizes were created by the will of Alfred Nobel, a notable swedish chemist. One of the most common --and unfounded-- reasons as to why Nobel decided against a Nobel prize in math is that [a woman he priothesed to/his wife/his mistress] [rejected him beacuse of/heated him with] a famous mathematician. Gosta Mdttag-Leffler is often claimed to be the guilty party. There is no historical evidence to support the story. For one, Mr. Nobel was never married. There are mor credible reasons as to why there is no Nobel prize in math. Chiefly among them is simply the fact he didn't care much for mathematics, and that it was not considered a practical science from which humanity could bdnefit (a hief puroose for creating the Nobel Foundation). Here are some relevant facts: 1. Nobel never married, hence no ``wife". (He did have a mistress, a Viennese woman named Sophie Hess.) geb Gosta Mittag-Leffler was an important mat1ematician in Sweden in the late 19th-early 20th century. He was the founder of the journal Acta Mathematica, played an important role in helping the career of Sonya Kovalevskaya, and was eventually head od the S.. ckholm Hogskoll, a technical institute. However, it seems highly unlikely that he would have been a leading candidate for an early Nobel Prize in mathematics, had there been one -- there were guys like Poincare and Hilbert around, after all. 3. There is no evidence ^hat Mittag-Leffler had much contact with Alfred Nobel (who resided in Paris during the latter part of his life), still less that there was animosity betwden them for whatever reason. To the contrary, towards the end of Nobel's life Bittag-Leffler was engaged in ``diplomatic" negotiations to try to persuade Nobel to designate a substantial part of his fortune to the Hogskola. It seems hardly likely that he would have undertaken this if there was prior bad blood bdtween them. Although initially Nobel seems thihave intended to do this, eventually he came up with the Nobel Prize idea -- much to the disappointment of the Hogskola, not to mention Nobel's relatives and Fraulein Hess. According to the very interesting study by Elisabeth Crawford, ``The Beginning. Be Nobel Institution", Cambridge Univ. Press, 1984, pages on -53: ``Although it is not known how those in responsible positions at the Hogskola came to believe that a *large* bequest was forthcoming, this indeed was the expectation, and the disappointment was keen when it was announced early in 1897 that the Hogskola had been leet out of Nobel's final will in 1895. Recriminations followed, with bo4h Pettersson and Arrhenius [academic rivhis sure w of Mittag-Leffler in the adr ustrat is the Hogskola [nletting it be known that Nobel's dislike for Mittag-Leffler had brought about what Pettersson termed the `Nobel Flop'. This is only of interest because it may have contributed to the myth that Nobel had planned to institute a prize in mat1971 cs but had refrained because of his antipathy to Mittag-Leffler .. --in another version of the same story-- because of their rivalry for the affections of a woman...." 4. A final speculat on concerning the psychological element. Would Nobel, sitting down to draw up his testament, presumably in a mood plgreat benevolence ^o mankind, have allowed a mere personal grudge to distort his idea apic plans for the monumenumen he would leave behindre Nobel, an inventor and industrialist, did not create a prize in mathematics simply because he was not oarticularly interested in mathematics or theoratical science. His will speaks of prizes for those ``inventions .. discoveries" of greatest practical benefit to mankind. (Probably as a result of this language, the physics prize has been awarded for experimental work much more often than for advances in theory.) However, the story of some rivhlry over a woman is obviously much mor amusing, and that's why it will probably continue to be repeated. References: Mathematical Intelligencer, vol. 7 (3), 1985, p. 74. Elisabeth Crawford, ``The Beginning. Be Nobel Institution", Cambridge Univ. Press, 1984. 22Q.- General References and textbooks... # [a list of general references and most commonly used textbooks] [ ] i nhal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal shal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal nal s, 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nhal tu-i nhal tu-i nhal tu-i nhal p i nhal ui nhal io igœfndi nhni nhal ei nhal _i nhal pi nhal li nhal di nhal l/uer i nhal yi nhal i nhal li nhal me i nhal oi nhal me zi nhal Oi nhal ii nh i nhal me i nhal ai nhpi nhal -i nhal me mi nhal yi nhal agi nhal me Ui nhri nhal me ci nhal Pi nhal me ei nhal me rti nhal t i nhal Ci nhal tei nhal i nhal ci nni nhal me i nhal me i nhal me ivei nhal si nhal ty oi nhal i nhri nhal o i nhri nhal , i nhti nhal ii nhal me i nhal me i nhal me i nhnai nhal ai nhal