Pythagorean Triples and Fermat's Last Theorem
Pythagorean Theorem says that the sum of the squares of the sides of a right
triangle equals the square of the hypotenuse. In symbols,
are a few examples which satisfy the Pythagorean Theorem:
Are there infinitely
many Pythagorean triples? The answer is "YES" for a trivial reason
since, for example, for any integer d,
new triples are not interesting, so we concentrate only on triples with no
common factors. Such triples are called primitive
are some other triples:
(20; 21; 29); (12; 35; 37); (9; 40;
41); (16; 63; 65); (28; 45; 53).
Pythagorean Triples Theorem:
will get every primitive Pythagorean triple (a; b; c) with a odd and b even by using the formulas:
where s > t ≥1 are chosen to be any odd integers
with no common factors.
since b is even, we could have
started with . Hence and, we conclude that and so that , and b = 2uv where u >
v ≥ 1, (u,v) = 1, and u and v have opposite parity.
the equation has infinitely many solutions, it is natural
investigate the situation where the exponent 2 is replaced by 3, and then 4,
and so on.
example, do the equations
solutions in nonzero integers a, b, c?
1637 Pierre de Fermat(1601-1665) showed that there is no solution for exponent
has no nontrivial solutions in
that where (a, b, c) have no common factor and say
b is even. Then,
u > v ≥1, (u, v) = 1; and u and v of opposite parity. In fact, v must be
even for if not and u is even, then from ,
we have (mod4) which is not possible. Since ,
then as above
and hence p, q, pq and are squares since p, q, pq and have no factors in common. Let and .
Then is a square and
= u < <
sets up an infinite descent chain of
squares of whole numbers of the form which is clearly impossible. (The above
argument is called the method of infinite descent and was invented by Fermat.)
To learn more:
Triples and Fermat's Last Theorem
Memorial University of Newfoundland