CHEZ FERMAT A.D. 1637' Erkka Maula and Eero Kasanen Abstract ...

140 ERKKA MAULA & EERO KASANEN
possible for Fermat. In case his premisses consisted of Props.
1-3 and Lemmas 1-3, he could first show (by Lemma 5) that, if
FLT fails, then the exponent p, an odd prime, is (a) one of the
primes pAr, ..• , p"'w when p I a or (13) one of the primes q"'1, ..• ,
q"f when p t abc. Next, given a,b,c he could determine, independently
of Lemma 5, the highest potency of the acute-angled
scalene (n-l)-potent triangle (a,b,c) starting from Prop. 2 and
Lemma 3. Finally, Fermat could try to show that p = (n-l)+1 is
none of the primes pAr, ••• , pAW, qAl, •.. , qAf eventually establishing
a contradiction. If that succeeds, then FLT is proven.
This is an arithmetical strategy, and it is well known that
Fermat's predilections were arithmetical.
It was stated earlier (following Del. 4) that the potency of an
acute-angled scalene triangle (a,b,c), where a > b > c > 0,
depends both on the perimeter a+b+c = 2h which must be even
because of parity, and on how far apart a,b,c are from one
another. Consider now all scalene integer';"'side triangles with the
even perimeter 2h. They can be illustrated as a subset of points
in a Pythagorean discrete-point equilateral triangle (Fig. 3).
py1;hagorean trianguJ.8J! nUD.bers generating a discrete··spac.
Inside the frB.II.e ABl the fo1l.owin& triangles:
~
l.:4.13.31
14.12.4
14,ll,5
14,10,6
~B:t~j~
g~:~?b1
E'
(12,11,7) (2)-potent
(12,10,8) (2)-potent
DETAIL
(u:;:0,9) (4)-potent
-for. these alone it
holds a,. b > c ,. 0 •••• • • •
In this eXB.II.p1e • • • • • • • •
B+b+C-O (mOd 6) .', ••••••,••••••••••
A~ ____ a_sb_~_c ________ ~B
o 0 0 0 0
o 0 p. 0
iY"
O (j
o 0 a
..a
o
a·b·e
C
A,~ ____ ~~~ ______ ~
h.!C;+b+C)
PQ·h-a
PR.h-b
ps·h-c
E