What is a Pythagorean triple? What do Pythagorean triples have to do with
Fermat's Last Theorem?
A Pythagorean triple is a set of three positive whole numbers
a2 + b2 = c2.
The smallest example is
32 + 42 = 9 + 16 = 25 = 52.
Sometimes we use the notation (a,b,c) to denote such a triple.
Notice that the greatest common divisor of the three numbers
a2 + b2 = c2 if and only if (da)2 + (db)2 = (dc)2.
Thus (a,b,c) is a Pythagorean triple if and only if (da,db,dc) is. For example, (6,8,10) and (9,12,15) are imprimitive Pythagorean triples.
Formulas for Primitive Pythagorean Triples and Their Derivation
Suppose that we start with a primitive Pythagorean triple (a,b,c). If any two of a,b,c shared a common divisor d, then, using the equation
a2 + b2 = c2,
we could see that the d2 would have to divide the square of
the remaining one, so that d would have to divide all three. This would
contradict the assumption that our triple was primitive. Thus no two of
Now notice that not all of a, b, and c can be odd, because if
(2x+1)2 + (2y+1)2 = (2z+1)2,
This implies that 4 is a divisor of 3, which is false. Now suppose
that c were even, and a and b odd, so
(2x+1)2 + (2y+1)2 = (2z)2,
This implies that 4 is a divisor of 2, which is also false. Thus either a or b must be even, and the other two odd. Let's say it is b that is even, with a and c odd. (If not, switch the meanings of a and b in what follows below.)
Now rewrite the equation in the form
b2 = c2 - a2,
Notice that since b is even and a and c are odd, b/2,
This gives us the product of two whole numbers,
r2 = (c + a)/2,
Furthermore, r > s, because
Now, solving the above three equations for a, b, and c, we find that, for primitive Pythagorean triples,
To see that the a, b, and c defined by these formulas do form a Pythagorean triple, just check the equation:
The above formulas for a, b, and c are the most general formulas for
primitive Pythagorean triples. To every pair of whole numbers
r = sqrt([c + a]/2),
Table of Small Primitive Pythagorean Triples
Here is a table of the first few primitive Pythagorean triples (Michael Somos provides a larger Pythagorean Triple Table on the Web):
Formulas for All Pythagorean Triples
To include all Pythagorean triples, both primitive and imprimitive, we
These formulas represent every Pythagorean triple. Given a Pythagorean
Triple, we can recover
d = GCD(a,b,c),
There is a one-to-one correspondence between Pythagorean triples and
sets of values of the three parameters
Perimeter, Area, Inradius, and Shortest Side
The perimeter P and area K of a Pythagorean triple triangle are given by
P = a + b + c = 2r(r + s)d,
The radius of the inscribed circle, or inradius, is always a whole number, and is given by the formula s(r-s)d.
To determine the shortest side, it will be a if
a < b,
Similarly, the shortest side will be b if
What About Higher Powers?
The equation an + bn = cn,
If we allow more terms, the equation
a3 + b3 + c3 = d3
does have solutions, such as (3,4,5,6). So does
a4 + b4 + c4 = d4,
the smallest being (95800,217519,414560,422481).
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