Math FAQ- update for 6 July 1993 1Q.- Fermat's Last Theorem, status of .. # 11Q.- There are three doors, The Monty Hall problem, Master Mind and other games .. # 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 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. ... (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 Sophie Germain criterion Second 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 between 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". Masser 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 N(x) 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 theorem, due to Mason that essentially says the ABC conjecture IS true for polynomials. The ABC conjecture also implies Szpiro's conjecture [and vice-versa] and Hall's conjecture. These results are all generally believed to be true. There is a generalization of the ABC conjecture [by Vojta] which is too technical to discuss but involves heights of points on non-singular algebraic varieties . Vojta's conjecture also implies Mordell's theorem [already known to be true]. There are also a number of inter-twined conjectures involving heights on elliptic curves that are related 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 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]. (c) Conjectures arising from some conjectured inequalities involving Chern classes and some other deep results/conjectures 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 people who DO understand this stuff is fairly small. The diophantine and elliptic curve conjectures all involve deep properties of integers. Until these conjecture were tied to FLT, FLT had been regarded by most mathematicians as an isolated problem; a curiosity. 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.Sompolski "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, Thirteen Lectures on Fermat's Last Theorem, Springer 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 11Q: There are three doors, and there is a car hidden behind one of them, Master Mind 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 for rec.puzzles, send to the address archive-request@questrel.com an email message consisting of the text send switch Also any other FAQ list can be obtained through anonymous ftp from rtfm.mit.edu. References American Mathematical Monthly, January 1992. For the game of Master 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 (I think), which always solved the 4 peg 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 Master Mind, J. Recreational Mathematics 9 (1976-77), 1-6. 19Q: What is the Erdos Number? Form an undirected graph where the vertices are academics, and an edge connects academic X to academic Y if X has written a paper with Y. 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 of Erdos have Erdos number 1. Einstein has Erdos number 2, since he wrote 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 Erdos? Paul 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 related 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?, American Mathematical Monthly v. 76 (1969), p. 791. 20Q: 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. 21Q.- Why is there no 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 proposed to/his wife/his mistress] [rejected him beacuse of/cheated him with] a famous mathematician. Gosta Mittag-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 more 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 benefit (a chief purpose 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.) 2. Gosta Mittag-Leffler was an important mathematician 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 of the Stockholm Hogskola, 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 that 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 between them for whatever reason. To the contrary, towards the end of Nobel's life Mittag-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 between them. Although initially Nobel seems to have 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 Beginnings of the Nobel Institution", Cambridge Univ. Press, 1984, pages 52-53: ``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 left out of Nobel's final will in 1895. Recriminations followed, with both Pettersson and Arrhenius [academic rivals of Mittag-Leffler in the administration of the Hogskola] letting 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 mathematics but had refrained because of his antipathy to Mittag-Leffler or --in another version of the same story-- because of their rivalry for the affections of a woman...." 4. A final speculation concerning the psychological element. Would Nobel, sitting down to draw up his testament, presumably in a mood of great benevolence to mankind, have allowed a mere personal grudge to distort his idealistic plans for the monument he would leave behind? Nobel, an inventor and industrialist, did not create a prize in mathematics simply because he was not particularly interested in mathematics or theoretical science. His will speaks of prizes for those ``inventions or 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 rivalry over a woman is obviously much more amusing, and that's why it will probably continue to be repeated. References: Mathematical Intelligencer, vol. 7 (3), 1985, p. 74. Elisabeth Crawford, ``The Beginnings of the Nobel Institution", Cambridge Univ. Press, 1984. 22Q.- General References and textbooks... # [a list of general references and most commonly used textbooks] [ ] -------------------------------------------------------------------------- Questions and Answers _Compiled_ by: Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca Department of Computer Science University of Waterloo Waterloo, Ontario Canada