Dear Claus-Peter,

 

I have reflected on our discussion. I begin to understand your position and , at least in some cases you are right. Let's consider the "famous" theorem of the Pythagorean triangle. Point of view of the ordinary indefinite descent: giv en the Pythagorean triple (a,b,c) and supposing ab/2=x^2, we find another tripl e (a',b',c') with 1<c'<c and a'b'/2=x'^2 and another such that 1<c''&l t;c'<c. Absurd because we would have an infinite descent in integers. Point of view of the reduction-descent: in the reduction still we have 1<c''& lt;c'<c, therefore we should reach the triples with a "small hypothenuse", b ut anyway greater than 1. For these triples ab/2 is not a square, while it should be. Absurd. 

In these cases, you are right: it is only questions of taste of a math ematician to pose the argument in a way or in the other way, but the logical eq uivalence is trivial, nothing to prove.

 

 But I am not sure that this is always the case. Please, let consider the demonstration by Gauss, I have referred in my book, pages 424-426 and the c onsiderations at pages 462-464. In the case of Gauss, it seems to me not so eas y and immediate to find a set of initial values to which apply the reducti on-descent.

 

After all, I have decided to put the demonstration of the Pythagorean trian gle, like we agreeded. It is more significant than the fourth-powers case of Fermat's last theorem. Substantially I have got this demonstration from my book, modifying what was necessary.

 

Even if, at the end, we conclude that reduction-descent and ordin ary indefinite descent are the same method, I think that our paper makes sense because we could speak of applications of the same method and at least our pape r could be definitive on the indefinite descent, affirmative and negative propo sitions, new principles, ecc. Arguments on which the ideas are not yet cle ar (also the Unguru-Fowler discussion shows this).

 

Exactely with regard to Unguru-Fowler I gave a quick reading, I will r ead more thoruoghtly and express my opinion. Anyway it is interesting material& nbsp;I did nok know and I thank you.

 

For Kuhn, I need to read the book onve again because I read it 15 years ago . If for you the answer is not urgent, I will do it.

But excuse me, Klaus-Peter, something is not clear to me: in this moment do you have some lections at the university; do you have some students?

 

Hear you soon,

 

ciao,

 

Paolo 

----- Original Message ----
From: Claus-Peter Wirth <cp@ags.uni-sb.de >
To: paolo bussotti <bussottipaolo@yahoo.com>
Sent: Tuesday, Ja nuary 9, 2007 6:12:05 PM
Subject: Re: Merry Xmas