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§3. Pythagoras, Diophantus, and Fermat 9

n
n
x + y = 1,
n
n
n
or in projective coordinates by w = x + y .
While F 2 has infinitely many rational points on it as given above, Fermat, in
1621, conjectured that the only rational points on F n for 3 ≤ n were the obvious
ones. This is called Fermat’s last theorem.
(3.3) Fermat’s Last Theorem. For 3 ≤ n, the only rational points on F n lie on the
x-axis and y-axis.
Fermat stated the theorem in the following form:
Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-
quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in
duas ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem
sane detexi. Hanc marginis exiguitas non caperet.

It is the last comment that has puzzled people for a long time. Proofs were given
for special values of n by many mathematicians: For n = 4 by Fermat using (3.1), for
n = 3 by Euler in 1770, for n = 5 by Legendre in 1825, and for n = 7byG.Lam´ e
in 1839. The conjecture of Fermat, that is, Fermat’s last theorem, had been checked
for all n up to a very large six-digit number, and Kummer proved it for all n a regular
prime. Only in 1983 as a solution to the more general Mordell conjecture was given
by Gerd Faltings, did we know that F n (Q) has at most finitely many points. We will
return to the Mordell conjecture in §6. Finally in 1995 through the effort of A. Wiles
and others can we say Fermat’s Last Theorem is established, see Chapter 18.
Again we return to a problem related to the unit circle. Recently J. Tunnell has
considered the problem of the existence of Pythagorean triples (a, b, c) of positive
rational numbers where the area A = (1/2)ab of the right triangle is given.
For example, for (3, 4, 5) the area is 6 and for (3/2, 20/3, 41/6) the area is 5. It
can be shown that there are no right triangle with rational sides and area 1, 2, 3, or 4.
Thus the problem is not as elementary as it would appear at first glance. We will see
that it reduces to the question of rational points on certain cubic curves.
Observe that if A is the area of the right rational triangle with sides (a, b, c),
2
then m A is the area of the rational right triangle (ma, mb, mc). Hence the question
reduces to the case of right rational triangles with square-free integer area A. Further,
we can order the triple so that a < b < c.
(3.4) Proposition. For a square-free natural number A there is a bijective corre-
spondence between the following three sets:
2
2
2
(1) Triples of strictly positive rational numbers (a, b, c) with a +b = c ,a < b <
c, and A = (1/2)ab.
(2) Rational numbers x such that x, x + A, and x − A are squares.
2
2
3
(3) Rational points (x, y) on the cubic curve y = x − A x such that x is a square
of a rational number, the denominator of x is even, and y > 0.