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From: Bill Dubuque on 26 Jun 2010 17:41
Archimedes Plutonium <plutonium.archimedes(a)gmail.com> wrote:
>
> The bad news is that Hardy & Moongold [7] and the editors of Mathematical
> Intelligencer chose to lift Archimedes Plutonium's work without reference
> or attribution on this subject and present it into Mathematical
> Intelligencer as their own original work without so much as even a
> recognition of all the work done by AP on this subject, for the
> issue of attribution is lacking in this article. Maybe the magazine
> feels that sci newsgroup posts are unworthy of referencing and that
> ideas and posts are free to lift from the sci newsgroups.

In fact it's quite common that authors fail to acknowledge discussions
on electronic forums. This has happened to me many times, e.g. see my
emails below for a striking case regarding my old Wronskian-based proof
of Mason's ABC theorem. Certainly your huge number of posts here on
Euclid's proof have helped make it widely known that Euclid's proof
was not a proof by contradiction and, moreover, that many authors
were not aware of such. A quick google search shows that Michael Hardy
participated in one of these early threads in 1994 [4], so one would
presume that he knew of such discussions here, esp. since he seems
to be a fairly active member since then (using many different email
addresses - which makes it a bit difficult to locate all of his posts).

So it seems a bit strange when Michael Hardy claims in the article
that he first learned Euclid's proof was not by contradiction only in
2007, from Jitse Niesen on the Citizendium web site (presumably [5]).
Also, it seems like a strange coincidence that one of his "students"
thought he discovered a way to turn Euclid's proof into a proof of the
twin-prime conjecture - as you often claimed here. Perhaps Michael's
memory for sources is fuzzy. Or perhaps his editor's did not like
references to newsgroups. In any case I just wanted to let you know
that you are not alone - it's happened to me and probably to many
other frequent posters. Below are said emails on one of my examples:

-------- excerpt of email from 14 Mar 2005 --------

Hi, I just noticed a reference to Noah Snyder's proof of Mason's ABC
theorem in your lecture [0]. I wonder if this is really any different
than the Wronskian viewpoint that I've pointed out since the mid 80's,
which has been mentioned in passing in various places online since at
least '96 e.g. [2] (and [1], one of many MathWorld pages based wholly
on one of my math-fun posts - some uncredited). Do you know how
I might obtain a copy of Snyder's article? (later: see [6])
[...]
Thanks so much for making Synder's paper available. As I suspected
Snyder's proof is essentially the same proof I gave over 20 years ago.
I've mentioned this online in many places, e.g. I wrote in 1996 [2]
to math-fun & sci.math

Mason's abc theorem may be viewed as a very special instance of a
Wronskian estimate: in Lang's proof the corresponding Wronskian
identity is c^3 W(a,b,c) = W(W(a,c),W(b,c)), thus if a,b,c are
linearly dependent then so are W(a,c),W(b,c); the sought bounds
follow upon multiplying the latter dependence relation through by
N0 = r(a) r(b) r(c), where r(x) = x/gcd(x,x').

This is *precisely* what Snyder does in his proof. The first hit on
Googling "Mason's theorem" is the MathWorld page [1], which refers to
my 1996 post [2] which mentions this proof. Thus I'm surprised that
Snyder and his mentors/editors didn't know this.

There are actually much deeper things one can do from this Wronskian
viewpoint - it is a fundamental approximation tool in differential
algebra. In the mid eighties I was working for the Macsyma group on
effective approaches for special functions using tools from differential
algebra, so I was quite familiar with Wronskian tools. Hence it was a
nice confluence of events that Mason's work happened around the same
time, since I immediately recognized the relationship. If I'm lucky
enough to re-obtain my math library I should dig up some of my notes on
this and write a letter to the editor since there are still some things
worthy of mention.

--Bill Dubuque

[0] http://www.fen.bilkent.edu.tr/~franz/ag05/ag-02.pdf
[1] http://mathworld.wolfram.com/MasonsTheorem.html
see below for a snapshot
[2] http://groups.google.com/group/sci.math/msg/4a53c1e94f1705ed
http://google.com/groups?selm=WGD.96Jul17041312(a)berne.ai.mit.edu
[4] http://groups.google.com/group/sci.math/msg/2c73f0dab34a188d
[5] http://locke.citizendium.org/wiki/Talk:Prime_number/Archive_2

[6] Noah Snyder, An Alternate Proof of Mason's Theorem,
Elemente der Mathematik, Vol. 55, Issue 3, 2000, pp. 93-94
http://dx.doi.org/10.1007/s000170050074

[7] Michael Hardy and Catherine Woodgold
Prime Simplicity, Math. Intelligencer, v.31 , 4, Dec. 2009, 44-52
http://dx.doi.org/10.1007/s00283-009-9064-8

-------- Mathworld page [1] excerpted from my math-fun post --------

Mason's theorem may be viewed as a very special case of a Wronskian estimate
(Chudnovsky and Chudnovsky 1984). The corresponding Wronskian identity in the
proof by Lang (1993) is c^3 W(a,b,c) = W(W(a,c),W(b,c)) so if a, b, and c
are linearly dependent, then so are W(a,c) and W(b,c). More powerful Wronskian
estimates with applications toward Diophantine approximation of solutions of
linear differential equations may be found in Chudnovsky and Chudnovsky (1984)
and Osgood (1985).

REFERENCES:
Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian Formalism for Linear
Differential Equations and Pade Approximations." Adv. Math. 53, 28-54, 1984.

Dubuque, W. "poly FLT, abc theorem, Wronskian formalism [was: Entire
solutions of f^2+g^2=1]." math-fun(a)cs.arizona.edu posting, Jul 17, 1996.

Lang, S. "Old and New Conjectured Diophantine Inequalities."
Bull. Amer. Math. Soc. 23, 37-75, 1990.

Lang, S. Algebra, 3rd ed. Reading, MA: Addison-Wesley, 1993.

Mason, R. C. Diophantine Equations over Functions Fields. Cambridge, England:
Cambridge University Press, 1984.

Osgood, C. F. "Sometimes Effective Thue-Siegel-Roth-Schmidt-Nevanlinna Bounds,
or Better." J. Number Th. 21, 347-389, 1985.

Stothers, W. W. "Polynomial Identities and Hauptmodulen."
Quart. J. Math. Oxford Ser. II 32, 349-370, 1981.

Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian Formalism for
Linear Differential Equations and Pade Approximations."
Adv. Math. 53, 28-54, 1984
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