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Jul 17, 1996, 3:00:00 AM7/17/96

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"Harold P. Boas" <bo...@math.tamu.edu> wrote to sci.math.research on 7/3/96:

:

:Robert Israel wrote:

:>

:> Alan Horwitz <al...@psu.edu> writes:

:> |> I am interested in all entire solutions f and g to f^2+g^2=1.

:> |> I remember seeing this somewhere, but I cannot recall where.

:>

:> I've also seen this before, in fact I recall assigning it as homework

:> to one of my classes, but I don't recall the source. The solutions are ...

:

:Robert B. Burckel gives some history about this problem in his

:comprehensive book An Introduction to Classical Complex Analysis,

:volume 1 (Academic Press, 1979). In Theorem 12.20, pages 433-435,

:he shows that the equation f^n+g^n=1 has no nonconstant entire

:solutions when the integer n exceeds 2; when n=2, the solution

:is as given by R. Israel in his post. ... (papers of Fred Gross)

Note that the rational function case of FLT follows trivially from

Mason's abc theorem, e.g. see Lang's Algebra, 3rd Ed. p. 195 for a

short elementary (high-school level) proof of both. Chebyshev also

gave a proof of FLT for poly's via the theory of integration in

finite terms, e.g. see p. 145 of Shanks' "Solved and Unsolved Problems

in Number Theory", or Ritt's "Integration in Finite Terms", p. 37.

The Chebyshev result is actually employed as a subroutine of Macsyma's

integration algorithm (implemented decades ago by Joel Moses). Via abc

a related result of Dwork is also easily proved: if A,B,C are fixed

poly's then all coprime poly solutions of A*X^a+B*Y^b+C*Z^c = 0

have bounded degrees provided 1/a+1/b+1/c < 1. Other applications

in both number and function fields may be found in Lang's survey [3].

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').

More powerful Wronskian estimates with applications toward

diophantine approximation of solutions of LDEs may be found in

the work of the Chudnovsky's [1] and C. Osgood [2]. References

to recent work may be found (as usual) by following MR citations

to these papers in the MathSci database.

I have not seen mention of this Wronskian view of Mason's abc theorem.

Although elementary, it deserves attention since it connects the abc

theorem with the general unified viewpoint of the Wronskian formalism

as proposed by the Chudnovsky's and others.

[1] Chudnovsky, D. V.; Chudnovsky, G. V.

The Wronskian formalism for linear differential equations and Pade

approximations. Adv. in Math. 53 (1984), no. 1, 28--54.

86i:11038 11J91 11J99 34A30 41A21

[2] Osgood, Charles F.

Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better.

J. Number Theory 21 (1985), no. 3, 347--389. 87f:11046 11J61 12H05

[3] Lang, Serge

Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc.

(N.S.) 23 (1990), no. 1, 37--75. 90k:11032 11D75 11-02 11D72 11J25

-Bill

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