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PYTHAGOREAN TRIPLES

A Pythagorean triple is defined as a set of three positive integers (a,b,c) where a < b < c, such that

                                                         a2 + b2 = c2.

The sides of a right triangle follows the Pythagorean Theorem,

          a2 + b2 = c2

where a and b are the lengths of the legs of the right triangle while c is the length of the hypothenuse.

Each Pythagorean triple forms the length of the sides of a right triangle, whose perimeter is P = a + b + c.

A right triangle with sides of lengths 3, 4 and 5 is a special right triangle in that all the sides have whole number lengths.  The three numbers 3, 4 and 5 forms a Pythagorean triplet or Pythagorean triple.

A Pythagorean triplet is a set of three whole numbers where the sum of the squares of the first two is equal to the square of the third number.  Below are examples of Pythagorean triplets:

 

3

4 5

5

12 13
7 24 25
9 40 41
11 60 61
     
     

One equation satisfying a Pythagorean Triplet A, B, C is

     Given A is odd, then

     B = (A2 - 1)/2

     C = (A2 + 1)/2

Another equation derived by Plato was

     (m2+1)2 = (m2-1)2 + (2m)2

where m is a natural number.  The above equation is called Plato's Formula.

Euclid has also another method, namely:

Given integers x and y,

      A = x2 - y2

      B = 2xy

      C = x2 + y2

Pythagorean triples are called primitive triples if a,b,c are coprime, that is, if their pairwise greatest common divisors gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. Because of their relationship through the Pythagorean theorem, a, b, and c are coprime if a and b are coprime (gcd(a,b) = 1).

 

More Mathematical Recreations

Enjoy these other topics in mathematics:

MARSENNE'S PRIME

PERFECT NUMBERS

PLATONIC SOLIDS 

MOBIUS STRIP 

PRIME NUMBERS

GOLDEN RATIO

PI

FERMAT'S LAST THEOREM

SPIRALS

SOPHIE GERMAIN PRIMES

EULER'S CONSTANT

FIBONACCI SERIES

EQUIANGULAR SPIRAL

GREGORIAN CALENDAR

SUDOKU

HARMONIC TRIANGLE

SUM OF SQUARES

FRACTALS

MANDELBROT SET

CHINESE NEW YEAR

EULER PATH

GOLDEN TRIANGLE

HAMILTONIAN CYCLE

FOUR COLOR THEOREM

CHINESE POSTMAN PROBLEM

CANTOR SET

AN UNPREDICTABLE EVENT

SAFE PRIMES

MAGIC SQUARES

PASCAL TRIANGLE

GOLDEN RECTANGLE

GOLDEN TRIANGLE

HYPERBOLA

FOUR SQUARE THEOREM

 

 

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