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

142 ERKKA MAULA & EERO KASANEN
Elementa IX.16-17 can be made use of. Although LemmB 4 and
LemmB 6 are at one's disposal here, no corresponding results can
be obtained for the other two Heronic (actually Archimedean, cf.
[7, vol. ü:322]) constituents (a-b+c), (a+b-c). Hence this is one
more cul-de-sac; we have not been able to find any promising
continuation in this direction.
We have, however, one more possible strategy of praof ta
offer.
An Ancient StrBtagem
We recall that aince Fermat himself had proven the case p(x,4) +
p(y,4) 1- p(z,4), it is sufficient to prove FLT for ail odd prime
exponents. By Prop. 4, it is sufficient to prove FLT for ail odd
exponents, say, of the form 2k+ 1. This gives rise ta a proof-idea,
perhaps the simplest of them all. If a > b > c are integer sides of
a (2k)-potent acute-angled scalene triangle and p(a,2k+l) =
p(b,2k+l) + p(c,2k+l), then (p(a,(2k+l)/2), p(b,(2k+l)/2),
p(c,(2k+l)/2» is a right-angled triangle, and conversely. The
exponents are arithmetical means of two consecutive integers
k+l,k (k ~ 1), and the sides of the right-angled triangle thus
geometric means of sides of an acute-angled triangle
(p(a,k),p(b,k),p(c,k» and of an obtuse-angled triangle
(p(a,k+l),p(b,k+l),p(c,k+l», which are themselves potencies of
the (2k)-potent acute-angled triangle (a,b,c). It suffices, therefore,
ta prove the following proposition.
Prop. 5. If a > b > c > 0 are pairwise relatively prime
integers and sides of an acute-angled triangle such that
p(a,k), p(b,k), p(c,k) are sides of an acute-angled triangle
and p(a,k+l), p(b,k+1), p(c,k+1) are sides of an obtuseangled
triangle and their geometric means are sides of a
right-angled triangle, then 2k is not an integer.
The rest of the formaI part of this essay pertains to that.
Proof-ideBS
Suppose the opposite: k, k+ 1 are two consecutive integers where
k ~
1, a > b > c > 0 are pair wise relatively prime integers and
sides of an acute-angled (2k)-potent scalene triangle such that
p(a,k), p(b,k), p(c,k) are sides of an acute-angled triangle,
p(a,k+1), p(b,k+1), p(c,k+1} are sides of an obtuse-angled triangle
and their geometric means p(a,(2k+1)/2), p(b,(2k+1)/2),
p(c,(2k+1)/2) are sides of a right-angled triangle.