One of the benefits of sending a manuscript to a journal, let alone its being published, is that one gets to notice the many errors and typos until then quite invisible. No doubt my writing this letter will expose yet more mistakes.
My `Remarks on Fermat's Last Theorem' is inter alia blighted by the following. Six of the thirteen [not `twelve'] books of the Arithmetica were found in a monastery in Germany by Johannes Müller (Regiomantus) in 1464 [I have not the slightest idea how I came to write `1570']. I am indebted to New Zealand's Garry Tee for drawing these errors to my attention. Garry also complains that Fermat did not claim to have proved that all Fermat numbers $F_n=2^{2^n}+1$ are prime, but rather speculated that they might be. Just so. I have changed the offending remark to `Just so, Euler found that Fermat had stumbled in suggesting that the sequence 3, 5, 17, 257, 65537, 4294967297, ..., might consist just of prime numbers.' That's just as much a blunder on Fermat's part since he had every business realising that 641|F5.
Other errors: John Coates is saddled with the title `Sadleirian Professor' [an `i' was elided in the Gazette], and of course it's Matthias Flach. Hans Lausch drily asked me just when Euler was in Konigsberg? I have bridged that illusion by omitting any place (Euler spent his mathematical life in St Petersberg interrupted by a long stint in Berlin).
The equation a^t+b^u=c^v has lots of seemingly nontrivial solutions unless one insists that a, b and c have no common factor. I slurred that qualification. The relevant remark now reads:
Darmon and
Granville (1993) suggest Fermat might have chosen a different generalisation.
Led by this, I suggest that if a, b, c
are relatively prime then
a^t+b^u=c^v
has no solution in integers greater than 1 if all of t,
u, v are at least
3. If one exponent is allowed to be $2$ things are different. For example,
in the cases
(t,u,v)=(3,3,2) and
(4,3,2) there are infinitely many solutions.
In general, if
1/t + 1/u + 1/v < 1
we have grounds for believing that there are
just finitely many solutions for which a and b have no common factor.
One of the large known solutions is
43^8 + 96,222^3 = 30,042,907^2.
All nine solutions known have one or
other of the exponents equal to two.
Garry Tee also complains about my writing `Diophantus' rather than `Diophantos'. This seems a Cebyshev matter to me. When W.E. (Bill) Smith first introduced me to Tschebysheff polynomials he pointed out that one of the charms of the name `Cebycev' was that could spell it virtually any way one liked without necessarily being wrong. Tschebotarev shares this happy property.
Corrected offprints of the manuscript are available by writing to alf@mpce.mq.edu.au enclosing a physical address.
Alf van der Poorten
Macquarie University
alf@mpce.mq.edu.au
Consider $$ I=\int_{-1}^1 dx\; p(x) \int_{-1}^1 p(y) \; {dy\over x-y} $$ where the y-integral is a Cauchy principal value. This looks like a symmetric quadratic form with an anti-symmetric kernel, and hence we might expect that I=0 for any function p(x). A formal but spurious ``proof" of this result follows simply by interchanging x and y, then interchanging the order of integration, yielding I=-I immediately. The problem is that it is not in general legitimate to interchange orders of Cauchy principal-value integrals.
Indeed, at least for some functions p(x), the conclusion that I=0 is demonstrably false, as the example $$ p(x)=\sqrt{1-x\over 1+x} $$ shows. Thus, using a well-known Hilbert transform for the y-integral, we obtain $$ I=\int_{-1}^1 dx\; p(x)\; . \; \pi =\pi^2\,. $$
It seems likely that I=0 for bounded p(x), but that I takes a finite non-zero value whenever there is an inverse square root singularity in p(x) at either end of the integration range. For example, if p(x) is bounded except at x=-1, where $$ p(x)\to{Q\over\sqrt{1+x}} $$ for some constant Q, then I=1/2 pi^2 Q^2.
I am not quite sure how to prove this, or to generalise or complete it. The result has applications in aerodynamics, when p(x) is the pressure on a thin airfoil. There is usually an inverse square root pressure singularity at the leading edge x=-1, and I measures what is called the ``leading-edge suction'' force.
E.O. Tuck
Applied Mathematics Department
The University of Adelaide
SymbMath (an abbreviation for Symbolic Mathematics) is a symbolic calculator that can solve symbolic math problems. SymbMath is a computer algebra system that can perform exact numeric, symbolic and graphic computation. It manipulates complicated formulas and returns answers in terms of symbols, formulas, exact numbers, table and graph.
SymbMath is an expert system that is able to learn from user's input. If the user only input one formula without writing any code, it will automatically learn many problems related to this formula (e.g. it learns many integrals involving an unknown function f(x) from one derivative f'(x)).
SymbMath is a symbolic, numeric and graphics computing environment where you can set up, run and document your calculation, draw your graph, and use external functions in the same way as standard functions since the external functions are auto-loaded.
SymbMath is a programming language in which you can define conditional, case, piecewise, recursion, multi-value functions and procedures, derivatives, intergrals and rules.
SymbMath is database where you can search your data. SymbMath is a multi-windowed editor in which you can copy-and-paste anywhere in a file and between files, even from the Help file. It runs on IBM PCs (8086) with 400 KB free memory under MS-DOS.
It can provide analytical and numeric answers for:
Also included are:
Its three versions (Shareware, Student, and Advanced) are available from the author. The shareware version is available from anonymous FTP sites (e.g. ftp.unsw.edu.au in /pub/UNSW/symbmath/sm33a.zip).
If you get the SymbMath on ZIP format (e.g. sm33a.zip), unzip it with parameter -d by
pkunzip -d sm33a c:\symbmath
Weiguang Huang and D.B. Hibbert
Dept. Analytical Chemistry, UNSW
w.huang@unsw.edu.au, s9300078@cumulus.csd.unsw.oz.au
Inspired by the letter of M.D. Hirschhorn ( Gazette vol. 21, p.148) who discussed the ``strange'' differential equation
which, by putting y= u^2, becomes
I thought that the following picture may be appreciated.
Missing picture
Solution curves of equation (1)
Equation (2) is linear and inhomogeneous, its general solution can be computed by elementary methods (see e.g. [1], pp. 147-148), and the solutions of (1) become
whenever the expression inside the parantheses is non-negative. One sees that bifurcations from the trivial solution u = 0 (whenever no Lipschitz condition is satisfied) can occur wherever u' >= 0 for u = 0; i.e. wherever cosx + sinx >= 0, hence in the intervals pi/4 <= x < 3pi/4, 7pi/4 <= x < 11pi/4, and so on.
Gerhard Wanner
Section de Mathematiques
Universite de Geneve
The travelling Questacon Maths (sic) Centre (which is part of the National Science and Technology Centre) is based on a set of mathematical tasks which originate from the Campbell Mathematics Centre, established in the ACT by Neville de Mestre (Bond University). The skills required by the tasks span most school age children (and are enjoyable for adults as well). Geometry, algebra, and rudimentary ideas of functions all play a part. An integral aspect of the Questacon Maths Centre is the ``Hands on Mathematics Workshops'' developed and run by the staff. There is also an extensive array of inovative mathematics resource material on display for teachers to peruse while the students enjoy the challenge and their success at the centre.
The Questacon Maths Centre spends one or two school terms at a given location in Australia before moving on. In term 1, 1995 the centre was in country Victoria (Ararat then Bendigo) delighting school children and teachers who rarely have the opportunity to experience a Mathematics (or Science) Centre. Next it will be in Darwin (Term 2, 1995) and then Perth (Terms 3 and 4, 1995). ANZIAM and the Australian Mathematical Society have each contributed $1,000 in 1994 to the Centre to assist with some refurbishment of ageing tasks and the development of some new tasks. The money has been well spent and greatly appreciated, and in 1995 ANZIAM and the Australian Mathematical Society will again each contribute $1,000.
Members of the Society are actively encouraged to visit the Centre when possible and see the fine work being done and the impressive popularity of `hands-on' tasks with students and teachers. It is advisable to contact Sandy Clugston (06-2702811) in the first instance and mention your contact with ANZIAM or the Australian Mathematical Society.
Rodney Weber
ADFA
Canberra