Fun with Mathematics

Pythagorean triples with one same number

All the numbers in the Pythagorean triple cannot be odd,

because Sum of two odd squares will be even and cannot

be an odd square.Knowing the odd-even nature of the two

numbers of the Pythagorean triple, the odd-even nature of

the other number can be predicted.The permitted and

forbidden set of triples in terms of its odd-even

characteristics are given below                                  

                 permitted                                forbidden

        x        y        z                            x      y       z

      even  even  even                     even  even  odd

      even  odd    odd                      even  odd   even

      odd   odd   even                      odd   odd    odd

There are many puzzles associated with Pythagorean

triples. One among them is to find out two or more set

of triples with one identical number in them. This

identical number may be either the smallest or the

biggest of the set.

Factorisation technique is advantageous to generate such

sets.Any number (N), composite or prime can be expressed

as the product of its two factors (x,y). For primes it will the

1 and the prime number itself.

                                  N = x y

The  number N can be expressed in term of its factors as

N = [(x+y)/2]^2 – [(x-y)/2]^2

If N happens to be a square number N = n^2, then

n^2 + [(x-y)/2]^2 = [ (x+y)/2]^2

By using different pairs of two even or two odd factors for n^2

Pythagorean triples with one identical number n can be formed.

For example,

64 = 4x16 = 2x 32

which give (8,6,10) and (8,15,17)

From

 81 = 3x27 = 1x81 we have

(9,12,15) and (9,40,41)

If the square number has many pair of such factors

(both even or  both odd) ,then we can get three or

 even more sets with an identical number in them. For example,

144 = 2x72 =4x36=6x24=8x18

It gives,(12,35,37),(12,16,20),(12,9,15),(12,5,17)

Another way of getting such set of triples with identical bigger

number is linear combination of numbers in the given

Pythagorean triples(x,y,z) or with another

Pythagorean triples (a,b,c)

(ax+by)^2 + (ay-bx)^2 = (a^2+b^2)(x^2+y^2)= (cz)^2

For example

      x   y   z       a      b      c 

      3   4   5       7    24     25  gives 117^2 + 44^2 = 125^2

      3   4   5      24    7      25  gives 100^2 + 75^2 = 125^2

      3   4   5      20   15     25  gives 120^2 + 35^2 + 125^2

                                                                         (reducible)

It is very useful to generate double or triple coincidence in

numeral relations where sum of two squares is equated

with sum of two other squares. It is a different puzzle

which we will see in the next blog.