Abstract:
As a result of our survey on primitive Pythagorean triples, we were able to prove the
following theorem:
All primitive Pythagorean triples can be generated by almost one parametera ,
satisfyinga > 2 +1. Furthermore, a is either an integer or of the form
h
a = g where g
and h (> 1) are relatively prime numbers.
The proof of the theorem can be briefly outlined as follows:
Taking z = y + p for some p ³ 1, z 2 = y 2 + x2 can be put into the form
2 2
1 1
+ =
+
y
x
y
p
If
p
x a = , then the above equation can be put into the form
( )2 2 2 1+b = 1+a b ........................................................................
(1),
where
2
1 = a 2 -1
b
. Then the above equation can be reduced into
2
2 2 2 2
2
1
2
1
1 a a a +
- =
- + .
In order to generate primitive triples, the above equation has to be multiplied by 4 if a is
even and h 4 if
h
a = g . Now we are able to generate all the primitive Pythagorean triples
if a satisfies the conditions of our theorem and
2
a 2 -1 is reduced to cancel 2 in the
denominator whenever necessary. The condition a > 2 +1 and a is either integer or
of the form = (h > 1)
h
a g with g and h are relatively prime odd be imposed after a
careful study of the equation . In conclusion, an algorithm can be developed to
determine p and y so that (( y + p), y, x) is a primitive Pythagorean triple in the order
x < y < y + p for given x. A new theorem on primitive Pythagorean triples is found and it
may be useful in understanding the Fermat’s Last Theorem.